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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | by replacing the position of by its right child, and takes the position of in the BST, as shown in fig. 2 part (d).
Following is a pseudocode for the deletion operation in a binary search tree.
The procedure deals with the 3 special cases mentioned above. Lines 2-3 deal with case 1; lines 4-5 deal with case 2 and lines 6-16 for case 3. The helper function is used within the deletion algorithm for the purpose of replacing the node with in the binary search tree . This procedure handles the deletion | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | (and substitution) of from the BST.
Traversal
A BST can be traversed through three basic algorithms: inorder, preorder, and postorder tree walks.
Inorder tree walk: Nodes from the left subtree get visited first, followed by the root node and right subtree.
Preorder tree walk: The root node gets visited first, followed by left and right subtrees.
Postorder tree walk: Nodes from the left subtree get visited first, followed by the right subtree, and finally the root.
Following is a recursive implementati | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | on of the tree walks.
Examples of applications
Sort
A binary search tree can be used in sorting algorithm implementation. The process involves inserting all the elements which are to be sorted and performing inorder traversal. This method is similar to that of quicksort where each node corresponds to a partitioning item that subdivides its descendants into smaller keys and larger keys.
Priority queue operations
Binary search trees are used in implementing priority queues, using the element or node's ke | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | y as priorities. Adding new elements to the queue follows the regular BST operation; but the removal operation depends on the type of priority queue:
If it's an ascending order priority queue, removal of an element with the lowest priority is done through leftward traversal of the BST i.e. .
On the other hand, if it's a descending order priority queue, removal of an element with the highest priority is done through rightward traversal of the BST i.e. .
Types
There are many types of binary search trees.
| [
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] |
4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree |
Self-balancing binary search trees modify the basic insertion and deletion operations of binary search trees, often using additional information on each node, in order to maintain logarithmic depth. These include two early structures of this type, AVL trees, which maintain an invariant that subtree heights differ by at most one, and red-black trees, which instead color nodes red or black and maintain an invariant on the number of colored nodes on each root-to-leaf path. These two types of trees are unified | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | in the WAVL tree.
The T-tree is a height-balanced binary search tree optimized to reduce storage space overhead which are used for in-memory databases.
Weight-balanced trees achieve (a worst-case logarithmic time bound for searching and) constant time bounds (in an amortized sense: summing over whole sequences of operations rather than analysing the time for each operation independently) for the update operations, but can be more flexible as part of recursive structures used in range searching. When ke | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | ys are inserted in a random order to a non-self-balancing binary tree, or drawn independently from a random distribution, without deletions, the resulting random binary search tree has both logarithmic expected depth, and logarithmic worst-case depth with high probability.
The treap (tree heap) is a self-balancing version of binary search trees that obtains the same behavior for worst-case operations, by assigning random priorities to each key and using the priorities to structure the tree as a Cartesian | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | tree, with higher priorities at each node than at its children.
Certain types of self-balancing binary search trees have been designed to take advantage of non-uniform access patterns by handling frequently requested keys more quickly. The performance of these online binary search trees can be analyzed by their competitive ratio, the maximum possible ratio of its running time on a sequence of access requests compared to the time of the best possible self-balancing binary search tree for the same access req | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | uest. Splay trees have been conjectured to have a constant competitive ratio, but this has not been proven. The geometry of binary search trees gives a way of reformulating these problems geometrically, in terms of augmenting point sets to avoid axis-parallel rectangles with only two diagonal vertices present in the augmented set. Another online binary search tree, the tango tree, is inspired by this geometric formulation, and has been proven to achieve an competitive ratio, while only using additional bi | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | ts of memory per node.
A binary search tree may be "degenerate", by having only left children at every node, or by having only right children at every node. When the resulting degenerate binary search tree contains nodes it has height of . The performance or time complexity of a lookup operation is essentially identical with that of a linear search i.e. , which is alike that of data structures like arrays or linked lists.
Performance comparisons
In regards to performance characteristics of binary search | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | trees, a study shows that treaps perform better on average case, while red–black tree was found to have the smallest number of performance variations.
Optimal binary search trees
Optimal binary search tree is a theoretical computer science problem which deals with constructing an "optimal" binary search trees that enables smallest possible search time for a given sequence of accesses. The computational cost required to maintain an "optimal" search tree can be justified if search is more dominant activity | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | in the tree than insertion or deletion.
Threaded binary trees
A threaded binary search tree is an accessorial version of a binary tree whose pointers—either left or right fields of a node—points to the inorder successor or inorder predecessor of the given nodes such that efficient utilization of the placeholders fields are performed. Threading is classified into two categories:
One-way threading: The left or right pointer field of the nodes, holds a reference to the inorder predecessor or inorder succes | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | sor, but not both.
Two-way threading: The left and right pointer fields hold the references to the inorder predecessor and inorder successor respectively.
See also
Binary search algorithm
Search tree
Join-based tree algorithms
Weight-balanced tree
Notes
References
Further reading
External links
Ben Pfaff: An Introduction to Binary Search Trees and Balanced Trees. (PDF; 1675 kB) 2004.
Binary Tree Visualizer (JavaScript animation of various BT-based data structures)
Articles with example C++ c | [
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4320 | https://en.wikipedia.org/wiki/Binary%20search%20tree | Binary search tree | ode
Articles with example Python (programming language) code
Binary trees
Data types
Search trees | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the and the . A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple (L, S, R), where L and R are binary trees or the empty set and S is a singleton set containing the root. Some authors allow the binary tree to be the empty set as well.
From a graph theory perspective, binary (and K-ary) trees as defined here are arborescences. A bin | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | ary tree may thus be also called a bifurcating arborescence—a term which appears in some very old programming books, before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of binary tree to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted. A binary tree is a special | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | case of an ordered K-ary tree, where K is 2.
In mathematics, what is termed binary tree can vary significantly from author to author. Some use the definition commonly used in computer science, but others define it as every non-leaf having exactly two children and don't necessarily order (as left/right) the children either.
In computing, binary trees are used in two very different ways:
First, as a means of accessing nodes based on some value or label associated with each node. Binary trees labelled this | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | way are used to implement binary search trees and binary heaps, and are used for efficient searching and sorting. The designation of non-root nodes as left or right child even when there is only one child present matters in some of these applications, in particular, it is significant in binary search trees. However, the arrangement of particular nodes into the tree is not part of the conceptual information. For example, in a normal binary search tree the placement of nodes depends almost entirely on the or | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | der in which they were added, and can be re-arranged (for example by balancing) without changing the meaning.
Second, as a representation of data with a relevant bifurcating structure. In such cases, the particular arrangement of nodes under and/or to the left or right of other nodes is part of the information (that is, changing it would change the meaning). Common examples occur with Huffman coding and cladograms. The everyday division of documents into chapters, sections, paragraphs, and so on is an analo | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | gous example with n-ary rather than binary trees.
Definitions
Recursive definition
To actually define a binary tree in general, we must allow for the possibility that only one of the children may be empty. An artifact, which in some textbooks is called an extended binary tree is needed for that purpose. An extended binary tree is thus recursively defined as:
the empty set is an extended binary tree
if T1 and T2 are extended binary trees, then denote by T1 • T2 the extended binary tree obtained by by | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | adding edges when these sub-trees are non-empty.
Another way of imagining this construction (and understanding the terminology) is to consider instead of the empty set a different type of node—for instance square nodes if the regular ones are circles.
Using graph theory concepts
A binary tree is a rooted tree that is also an ordered tree (a.k.a. plane tree) in which every node has at most two children. A rooted tree naturally imparts a notion of levels (distance from the root), thus for every node a not | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | ion of children may be defined as the nodes connected to it a level below. Ordering of these children (e.g., by drawing them on a plane) makes it possible to distinguish a left child from a right child. But this still doesn't distinguish between a node with left but not a right child from a one with right but no left child.
The necessary distinction can be made by first partitioning the edges, i.e., defining the binary tree as triplet (V, E1, E2), where (V, E1 ∪ E2) is a rooted tree (equivalently arboresce | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | nce) and E1 ∩ E2 is empty, and also requiring that for all j ∈ { 1, 2 } every node has at most one Ej child. A more informal way of making the distinction is to say, quoting the Encyclopedia of Mathematics, that "every node has a left child, a right child, neither, or both" and to specify that these "are all different" binary trees.
Types of binary trees
Tree terminology is not well-standardized and so varies in the literature.
A binary tree has a root node and every node has at most two children.
A | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | binary tree (sometimes referred to as a proper or plane binary tree) is a tree in which every node has either 0 or 2 children. Another way of defining a full binary tree is a recursive definition. A full binary tree is either:
A single vertex.
A tree whose root node has two subtrees, both of which are full binary trees.
In a binary tree every level, except possibly the last, is completely filled, and all nodes in the last level are as far left as possible. It can have between 1 and 2h nodes at the last | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | level h. An alternative definition is a perfect tree whose rightmost leaves (perhaps all) have been removed. Some authors use the term complete to refer instead to a perfect binary tree as defined below, in which case they call this type of tree (with a possibly not filled last level) an almost complete binary tree or nearly complete binary tree. A complete binary tree can be efficiently represented using an array.
A binary tree is a binary tree in which all interior nodes have two children and all leave | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | s have the same depth or same level. An example of a perfect binary tree is the (non-incestuous) ancestry chart of a person to a given depth, as each person has exactly two biological parents (one mother and one father). Provided the ancestry chart always displays the mother and the father on the same side for a given node, their sex can be seen as an analogy of left and right children, children being understood here as an algorithmic term. A perfect tree is therefore always complete but a complete tree is | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | not necessarily perfect.
In the infinite complete binary tree, every node has two children (and so the set of levels is countably infinite). The set of all nodes is countably infinite, but the set of all infinite paths from the root is uncountable, having the cardinality of the continuum. That's because these paths correspond by an order-preserving bijection to the points of the Cantor set, or (using the example of a Stern–Brocot tree) to the set of positive irrational numbers.
A balanced binary tree is | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | a binary tree structure in which the left and right subtrees of every node differ in height by no more than 1. One may also consider binary trees where no leaf is much farther away from the root than any other leaf. (Different balancing schemes allow different definitions of "much farther".<ref>Paul E. Black (ed.), entry for data structure in Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. 15 December 2004. Online version Accessed 2010-12-19.</ref>)
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | enerate (or pathological) tree is where each parent node has only one associated child node. This means that the tree will behave like a linked list data structure.
Properties of binary trees
The number of nodes in a full binary tree is at least and at most , where is the height of the tree. A tree consisting of only a root node has a height of 0.
The number of leaf nodes in a perfect binary tree, is because the number of non-leaf (a.k.a. internal) nodes .
This means that a full binary tree with | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | leaves has nodes.
In a balanced full binary tree, (see ceiling function).
In a perfect full binary tree, thus .
The number of null links (i.e., absent children of the nodes) in a binary tree of n nodes is (n+1).
The number of internal nodes in a complete binary tree of n nodes is .
For any non-empty binary tree with n0 leaf nodes and n2 nodes of degree 2, n0 = n2 + 1.
Combinatorics
In combinatorics one considers the problem of counting the number of full binary trees of a given size. Here the | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | trees have no values attached to their nodes (this would just multiply the number of possible trees by an easily determined factor), and trees are distinguished only by their structure; however, the left and right child of any node are distinguished (if they are different trees, then interchanging them will produce a tree distinct from the original one). The size of the tree is taken to be the number n of internal nodes (those with two children); the other nodes are leaf nodes and there are of them. The nu | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | mber of such binary trees of size n is equal to the number of ways of fully parenthesizing a string of symbols (representing leaves) separated by n binary operators (representing internal nodes), to determine the argument subexpressions of each operator. For instance for one has to parenthesize a string like , which is possible in five ways:
The correspondence to binary trees should be obvious, and the addition of redundant parentheses (around an already parenthesized expression or around the full expre | [
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0.028009018,
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0.008721171,
0.018281814,
0.047703274,
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0.06646478,
0.026356725,
0.08010952,
0.02118665,
0.020493753,
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0.010653287,
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0.0055564987,
0.027502669,
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0.030674005,
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0.034858033,
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0.11544725,
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0.025037557,
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0.037123274,
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0.031047102,
0.028408766,
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-0.034858033,
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0.014390931,
0.049488816,
0.09956393,
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0.00015427779,
0.055005338,
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0.017442342,
0.020746926,
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0.01106636,
0.028222216,
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0.028861813,
0.07627194,
0.0321664,
0.019014684,
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0.032859296,
0.021199975,
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0.024224736,
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0.032273,
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0.024744408,
0.07877703,
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0.012245617,
0.035284434,
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0.08783799,
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0.008467997,
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0.0051967255,
0.011599357,
0.014004509,
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0.06753078,
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0.06097491,
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0.009967052,
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0.04399894,
0.01055335,
0.03363214,
0.042906296,
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0.004064106,
0.030833904,
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0.024144785,
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0.026623223,
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0.057457127,
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0.03768292,
0.063426696,
0.016536247,
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0.0637465,
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0.06822368,
0.043679144,
0.053566247,
0.032352947,
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0.041014157,
0.027142897,
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0.018095264,
0.04482509,
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0.0033095812,
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0.0045704534,
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0.039281912,
0.0076018763,
0.030407505,
0.028328815,
0.017668866,
0.052180454,
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0.021453148,
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0.041680403,
0.036483675,
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0.02313209,
0.057084028,
0.11960463,
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0.012771952,
0.029128311,
0.08650549,
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0.17429018,
0.029128311,
0.05457894,
0.047863174,
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0.023785012,
0.0054265806,
0.09881773,
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0.1471073,
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0.049248967,
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0.0065691937,
0.03155345,
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0.108678184,
0.0012317238,
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0.024091486,
0.0076351888,
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0.073393755,
0.0213732,
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0.021639697,
0.0102069015,
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0.023185391,
0.023984887,
0.00249676,
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0.07062217,
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] |
4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | ssion) is disallowed (or at least not counted as producing a new possibility).
There is a unique binary tree of size 0 (consisting of a single leaf), and any other binary tree is characterized by the pair of its left and right children; if these have sizes i and j respectively, the full tree has size . Therefore, the number of binary trees of size n has the following recursive description , and for any positive integer n. It follows that is the Catalan number of index n.
The above parenthesized strings | [
-0.025394375,
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0.06809276,
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0.0019691004,
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0.03720229,
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0.010591019,
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0.04201639,
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0.048996836,
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0.032334704,
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0.04474438,
0.041508235,
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0.044771127,
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0.0034969083,
0.012998069,
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0.07531391,
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0.033163797,
0.018627891,
0.022907091,
0.037229035,
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0.0005048118,
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0.004262484,
0.060336716,
0.06327867,
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0.09815414,
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0.015030689,
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0.07825586,
0.04094659,
0.17469832,
0.045519985,
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0.038673267,
0.041722197,
0.026731625,
0.04586767,
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0.019898279,
0.036079,
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0.015792921,
0.03832558,
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0.12987371,
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0.039529108,
0.019777926,
0.0042457683,
0.030810237,
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0.042631526,
0.015298138,
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0.006432172,
0.00013967993,
-0.033244032,
-0.018601146,
-0.0694835,
-0.027574092,
-0.010102923,
-0.002208134,
-0.017638326,
0.047766566,
0.0641345
] |
4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | should not be confused with the set of words of length 2n in the Dyck language, which consist only of parentheses in such a way that they are properly balanced. The number of such strings satisfies the same recursive description (each Dyck word of length 2n is determined by the Dyck subword enclosed by the initial '(' and its matching ')' together with the Dyck subword remaining after that closing parenthesis, whose lengths 2i and 2j satisfy ); this number is therefore also the Catalan number . So there ar | [
-0.0679644,
-0.006762993,
-0.06250584,
-0.03366111,
-0.0423841,
-0.01212121,
0.010442168,
0.015184959,
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0.054853156,
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0.10483643,
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0.019546455,
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0.016174993,
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0.08749747,
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0.031975377,
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0.018489528,
0.030985344,
0.024737436,
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0.045514744,
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0.03406247,
0.017713556,
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0.024670541,
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0.018195193,
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0.0038464104,
0.051214118,
0.028363096,
0.37824598,
0.019787272,
0.01200749,
0.007980466,
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0.023720646,
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0.010709745,
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0.03216268,
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0.03735366,
0.021687066,
0.05359555,
0.026008423,
0.013392198,
0.011960664,
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0.050759237,
0.015064551,
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0.01846277,
0.017138267,
0.015091307,
0.014984277,
0.010582645,
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0.018489528,
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0.037246633,
0.0025035115,
0.02140611,
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0.009485583,
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0.024576891,
0.04578232,
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0.04984948,
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0.045809075,
0.039387245,
0.0346779,
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0.03285838,
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0.07219211,
0.02107164,
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0.07754363,
0.023332661,
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0.030637495,
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0.020255532,
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0.035400353,
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0.037433937,
0.015706733,
0.029834766,
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0.049180537,
0.055869948,
0.014863868,
0.032457016,
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0.021325838,
0.008100875,
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0.057261344,
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0.03850424,
0.029433401,
0.015305369,
0.039467517,
0.026837911,
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0.023921328,
0.038370453,
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0.025874635,
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0.005020402,
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] |
4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | e also five Dyck words of length 6:
These Dyck words do not correspond to binary trees in the same way. Instead, they are related by the following recursively defined bijection: the Dyck word equal to the empty string corresponds to the binary tree of size 0 with only one leaf. Any other Dyck word can be written as (), where , are themselves (possibly empty) Dyck words and where the two written parentheses are matched. The bijection is then defined by letting the words and correspond to the binary tree | [
-0.031670377,
0.00007070776,
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0.01972359,
0.0043107644
] |
4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | s that are the left and right children of the root.
A bijective correspondence can also be defined as follows: enclose the Dyck word in an extra pair of parentheses, so that the result can be interpreted as a Lisp list expression (with the empty list () as only occurring atom); then the dotted-pair expression for that proper list is a fully parenthesized expression (with NIL as symbol and '.' as operator) describing the corresponding binary tree (which is, in fact, the internal representation of the proper | [
-0.037718542,
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | list).
The ability to represent binary trees as strings of symbols and parentheses implies that binary trees can represent the elements of a free magma on a singleton set.
Methods for storing binary trees
Binary trees can be constructed from programming language primitives in several ways.
Nodes and references
In a language with records and references, binary trees are typically constructed by having a tree node structure which contains some data and references to its left child and its right child | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | . Sometimes it also contains a reference to its unique parent. If a node has fewer than two children, some of the child pointers may be set to a special null value, or to a special sentinel node.
This method of storing binary trees wastes a fair bit of memory, as the pointers will be null (or point to the sentinel) more than half the time; a more conservative representation alternative is threaded binary tree.
In languages with tagged unions such as ML, a tree node is often a tagged union of two types of | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | nodes, one of which is a 3-tuple of data, left child, and right child, and the other of which is a "leaf" node, which contains no data and functions much like the null value in a language with pointers. For example, the following line of code in OCaml (an ML dialect) defines a binary tree that stores a character in each node.
type chr_tree = Empty | Node of char * chr_tree * chr_tree
Arrays
Binary trees can also be stored in breadth-first order as an implicit data structure in arrays, and if the tree i | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | s a complete binary tree, this method wastes no space. In this compact arrangement, if a node has an index i, its children are found at indices (for the left child) and (for the right), while its parent (if any) is found at index (assuming the root has index zero). Alternatively, with a 1-indexed array, the implementation is simplified with children found at and , and parent found at . This method benefits from more compact storage and better locality of reference, particularly during a preorder travers | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | al. However, it is expensive to grow and wastes space proportional to 2h - n for a tree of depth h with n nodes.
This method of storage is often used for binary heaps.
Encodings
Succinct encodings
A succinct data structure is one which occupies close to minimum possible space, as established by information theoretical lower bounds. The number of different binary trees on nodes is , the th Catalan number (assuming we view trees with identical structure as identical). For large , this is about ; thus | [
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0.36507416,
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0.0054110726,
0.0057109473,
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0.01735275,
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0.015940007,
0.028654696,
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0.0597617,
0.058588855,
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0.032786302,
0.038623866,
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0.06445307,
0.08998907,
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0.040782962,
0.004034981,
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0.0011761753,
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0.026375648,
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0.061094478,
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0.00966263,
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0.056936212,
0.009995824,
0.10091784,
0.026055781,
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0.016713018,
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0.031107005,
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0.0689312,
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0.008756341,
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0.07580833,
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0.09846553,
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0.023723422,
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0.008842972,
0.036891256,
0.002492292,
0.025242787,
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0.0042715487,
0.0033136157,
0.034252357,
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0.017406061,
0.06226732,
0.054350626,
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0.035531823,
0.086790405,
0.027241953,
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0.023363573,
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0.0631203,
0.047793366,
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0.05672297,
0.023350244,
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0.09713276,
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0.03739771,
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0.0226572,
0.009169502,
0.01264805,
0.047873333,
0.022950413,
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0.025416048,
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0.051018685
] |
4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | we need at least about bits to encode it. A succinct binary tree therefore would occupy bits.
One simple representation which meets this bound is to visit the nodes of the tree in preorder, outputting "1" for an internal node and "0" for a leaf. If the tree contains data, we can simply simultaneously store it in a consecutive array in preorder. This function accomplishes this:
function EncodeSuccinct(node n, bitstring structure, array data) {
if n = nil then
append 0 to structure;
e | [
-0.06690423,
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0.038863096,
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0.03819672,
0.033078946,
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0.026735038,
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0.052510493,
0.0041115447,
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0.0593875,
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0.075220615,
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0.035611175,
0.050671294,
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0.007136896,
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0.0068570175,
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0.032919016,
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0.023016656,
0.008482977,
0.043421112,
0.024056204,
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0.051737495,
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0.06519831,
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0.01763233,
0.023176586,
0.01649949,
0.007150223,
0.034465007,
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-0.040862225,
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0.02007127,
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0.054456312,
0.005540923,
0.008709545,
-0.032199327,
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0.014700272,
0.03803679,
-0.0010303851,
0.010755322,
0.008776182,
-0.052163977,
0.025922058,
-0.03238591,
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-0.016579455,
0.020377804,
-0.029133994,
0.006780384,
0.033585392,
0.06279935,
-0.038543236,
0.071328975,
-0.01649949,
-0.03321222,
0.07676661,
-0.07639344,
0.049178608,
0.002990366,
-0.015033461,
0.009802403,
0.025348974,
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0.023603067,
0.037983477,
0.04939185,
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0.009322612,
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0.040942192,
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0.0008937779,
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0.035317972,
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0.057788197,
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0.051630877,
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0.00868289,
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0.16408862,
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0.055176,
0.015366649,
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0.02710821,
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0.005687526,
0.025242353,
0.011201794,
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0.0746342,
0.047792546,
0.02585542,
0.0001875226,
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0.02104418,
0.00055850705,
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0.0009862377,
0.027068226,
0.0111818025,
0.008876139,
-0.0053343466,
0.051977392,
0.06807706,
0.019644788,
-0.0207643,
-0.016526144,
-0.0147269275,
0.052857008,
0.042808045,
0.018285379,
-0.035797764,
-0.05896102,
-0.03872982,
-0.012781107,
0.04907199,
-0.0105620725,
-0.016566128,
0.014740255,
0.010921916,
0.0169393,
0.048272338,
-0.005404316,
-0.071648836,
0.097610876,
0.09174676,
-0.0520307,
-0.017125884,
0.003361871,
-0.0331056,
0.027348105,
0.042754736,
0.10374154,
0.016379543,
0.008369693,
0.0625328,
0.056562062,
0.028227722,
0.029320579,
0.011874835,
-0.03982268,
0.025388956,
0.010761986,
-0.012114731,
0.02417615,
-0.030733299,
-0.26932284,
-0.004254816,
0.013780672,
0.011061855,
0.053043593,
-0.02039113,
0.02428277,
-0.04440735,
-0.0417685,
-0.00032714938,
-0.02412284,
0.028387653,
0.07644675,
-0.05528262,
0.007736635,
0.006473851,
0.07985859,
0.012634505,
-0.04797913,
-0.03590438,
0.05954743,
0.028360996,
0.2018322,
0.04803244,
-0.04840561,
-0.0035151376,
-0.027348105,
0.028760822,
0.035317972,
0.010202229,
0.0039516143,
0.023842962,
0.10123596,
-0.020644354,
-0.046966236,
0.030759953,
0.0189251,
0.041848462,
0.05805475,
0.019591479,
-0.030040266,
0.0056408797,
-0.012747789,
-0.009402577,
0.055975653,
0.018045483,
0.046193242,
0.0072501795,
0.0061273347,
-0.013180933,
-0.027214829,
0.05789482,
-0.002587208,
-0.08284397,
-0.007183542,
0.031986088,
-0.029986957,
-0.02212371,
-0.10198231,
-0.006240619,
0.0041515273,
-0.027041571,
0.011674922,
0.019738082,
-0.019458203
] |
4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | lse
append 1 to structure;
append n.data to data;
EncodeSuccinct(n.left, structure, data);
EncodeSuccinct(n.right, structure, data);
}
The string structure has only bits in the end, where is the number of (internal) nodes; we don't even have to store its length. To show that no information is lost, we can convert the output back to the original tree like this:
function DecodeSuccinct(bitstring structure, array data) {
remove first bit of structure and put it in | [
-0.030389553,
-0.021373898,
-0.0062057544,
-0.065413445,
0.032147408,
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-0.030895602,
0.028258823,
0.013849753,
0.02354458,
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0.022798823,
0.002348798,
0.017978044,
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0.021200774,
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0.026421066,
0.10115646,
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0.02844526,
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0.040217552,
0.050791305,
0.0006075913,
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0.023371458,
0.0070247543,
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0.01077351,
0.0070913397,
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0.05127072,
0.05121745,
0.026088139,
0.002117414,
0.004930645,
0.016126972,
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0.033106234,
0.01954946,
-0.027752774,
0.0020874506,
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0.045064963,
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0.0616314,
0.005812901,
0.04511823,
-0.0192698,
0.00509045,
0.029164383,
0.013430265,
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0.085016176,
0.031614725,
0.033665553,
0.007970266,
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0.091035485,
0.017778289,
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0.059660472,
0.05148379,
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0.028871408,
0.017471995,
0.013703264,
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0.03281326,
0.07478866,
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-0.010001119,
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0.009455119,
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0.05545228,
0.013982923,
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0.0826191,
0.014582192,
0.33942547,
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0.046796184,
-0.0026134748,
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-0.0003603932,
0.0118189,
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-0.014129411,
-0.006065925,
0.01699258,
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-0.026181359,
-0.013450241,
0.024756433,
0.02831209,
-0.020881165,
0.0064388034,
0.015008338,
0.02058819,
-0.038406428,
-0.03539677,
0.06253696,
0.043786526,
-0.00076032145,
0.04008438,
0.08112759,
0.091408364,
0.046290133,
0.016553117,
-0.045704182,
-0.034864087,
-0.05212301,
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0.039258722,
0.0067650713,
0.020668093,
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0.008982363,
0.0141160935,
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0.113461435,
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0.00034000142,
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0.007071364,
0.047568575,
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0.008409729,
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-0.02303853,
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-0.044692084,
0.050604865,
0.0032826576,
0.07047393,
0.022439262,
-0.0341716,
0.035050526,
0.07196544,
0.0626435,
-0.09551002,
0.002718347,
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0.023744335,
0.012804363,
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0.024183799,
0.07105988,
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0.0035822918,
0.00631562,
0.0109000215,
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | b if b = 1 then
create a new node n remove first element of data and put it in n.data
n.left = DecodeSuccinct(structure, data)
n.right = DecodeSuccinct(structure, data)
return n
else
return nil
}
More sophisticated succinct representations allow not only compact storage of trees but even useful operations on those trees directly while they're still in their succinct form.
Encoding general trees as binary trees
There is a one-to-one mapping | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | between general ordered trees and binary trees, which in particular is used by Lisp to represent general ordered trees as binary trees. To convert a general ordered tree to a binary tree, we only need to represent the general tree in left-child right-sibling way. The result of this representation will automatically be a binary tree if viewed from a different perspective. Each node N in the ordered tree corresponds to a node N' in the binary tree; the left child of N' is the node corresponding to the first | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | child of N, and the right child of N' is the node corresponding to N 's next sibling --- that is, the next node in order among the children of the parent of N. This binary tree representation of a general order tree is sometimes also referred to as a left-child right-sibling binary tree (also known as LCRS tree, doubly chained tree, filial-heir chain).
One way of thinking about this is that each node's children are in a linked list, chained together with their right fields, and the node only has a pointe | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | r to the beginning or head of this list, through its left field.
For example, in the tree on the left, A has the 6 children {B,C,D,E,F,G}. It can be converted into the binary tree on the right.
The binary tree can be thought of as the original tree tilted sideways, with the black left edges representing first child and the blue right edges representing next sibling. The leaves of the tree on the left would be written in Lisp as:
(((N O) I J) C D ((P) (Q)) F (M))
which would be implemented in memory as | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | the binary tree on the right, without any letters on those nodes that have a left child.
Common operations
There are a variety of different operations that can be performed on binary trees. Some are mutator operations, while others simply return useful information about the tree.
Insertion
Nodes can be inserted into binary trees in between two other nodes or added after a leaf node. In binary trees, a node that is inserted is specified as to whose child it will be.
Leaf nodes
To add a new node af | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | ter leaf node A, A assigns the new node as one of its children and the new node assigns node A as its parent.
Internal nodes
Insertion on internal nodes is slightly more complex than on leaf nodes. Say that the internal node is node A and that node B is the child of A. (If the insertion is to insert a right child, then B is the right child of A, and similarly with a left child insertion.) A assigns its child to the new node and the new node assigns its parent to A. Then the new node assigns its child to | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | B and B assigns its parent as the new node.
Deletion
Deletion is the process whereby a node is removed from the tree. Only certain nodes in a binary tree can be removed unambiguously.
Node with zero or one children
Suppose that the node to delete is node A. If A has no children, deletion is accomplished by setting the child of A's parent to null. If A has one child, set the parent of A's child to A's parent and set the child of A's parent to A's child.
Node with two children
In a binary tree, a | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | node with two children cannot be deleted unambiguously. However, in certain binary trees (including binary search trees) these nodes can be deleted, though with a rearrangement of the tree structure.
Traversal
Pre-order, in-order, and post-order traversal visit each node in a tree by recursively visiting each node in the left and right subtrees of the root.
Depth-first order
In depth-first order, we always attempt to visit the node farthest from the root node that we can, but with the caveat that it | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | must be a child of a node we have already visited. Unlike a depth-first search on graphs, there is no need to remember all the nodes we have visited, because a tree cannot contain cycles. Pre-order is a special case of this. See depth-first search for more information.
Breadth-first order
Contrasting with depth-first order is breadth-first order, which always attempts to visit the node closest to the root that it has not already visited. See breadth-first search for more information. Also called a level- | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | order traversal.
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | ere are no more. The rightmost bit indicates the final traversal from the desired node's parent to the node itself. There is a time-space trade-off between iterating a complete binary tree this way versus each node having pointer/s to its sibling/s.
See also
References
Citations
Bibliography
Donald Knuth. The Art of Computer Programming vol 1. Fundamental Algorithms'', Third Edition. Addison-Wesley, 1997. . Section 2.3, especially subsections 2.3.1–2.3.2 (pp. 318–348).
External links
binary | [
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4321 | https://en.wikipedia.org/wiki/Binary%20tree | Binary tree | trees entry in the FindStat database
Binary Tree Proof by Induction
Balanced binary search tree on array How to create bottom-up an Ahnentafel list, or a balanced binary search tree on array
Binary trees and Implementation of the same with working code examples
Binary Tree JavaScript Implementation with source code | [
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4322 | https://en.wikipedia.org/wiki/Borel%20measure | Borel measure | In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
Formal definition
Let be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of ; this is known as the σ-algebra of Borel sets. A Borel measure is any measure defined on the σ-algebra of Borel sets. A few authors requ | [
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4322 | https://en.wikipedia.org/wiki/Borel%20measure | Borel measure | ire in addition that is locally finite, meaning that for every compact set . If a Borel measure is both inner regular and outer regular, it is called a regular Borel measure. If is both inner regular, outer regular, and locally finite, it is called a Radon measure.
On the real line
The real line with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, is the smallest σ-algebra that contains the open intervals of . While there are many Bor | [
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4322 | https://en.wikipedia.org/wiki/Borel%20measure | Borel measure | el measures μ, the choice of Borel measure that assigns for every half-open interval is sometimes called "the" Borel measure on . This measure turns out to be the restriction to the Borel σ-algebra of the Lebesgue measure , which is a complete measure and is defined on the Lebesgue σ-algebra. The Lebesgue σ-algebra is actually the completion of the Borel σ-algebra, which means that it is the smallest σ-algebra that contains all the Borel sets and has a complete measure on it. Also, the Borel measure and t | [
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4322 | https://en.wikipedia.org/wiki/Borel%20measure | Borel measure | he Lebesgue measure coincide on the Borel sets (i.e., for every Borel measurable set, where is the Borel measure described above).
Product spaces
If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets of their product coincides with the product of the sets of Borel subsets of X and Y. That is, the Borel functor
from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.
Applications
Lebesgue–Stieltje | [
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4322 | https://en.wikipedia.org/wiki/Borel%20measure | Borel measure | s integral
The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind.
Laplace transform
One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral
An important special | [
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] |
4322 | https://en.wikipedia.org/wiki/Borel%20measure | Borel measure | case is where μ is a probability measure or, even more specifically, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a distribution function f. In that case, to avoid potential confusion, one often writes
where the lower limit of 0− is shorthand notation for
This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necess | [
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-0.008434706,
-0.030807331,
0.050981335,
-0.032992627,
0.025850443,
-0.010919813,
0.07280764,
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-0.0128386095,
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0.09348799,
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0.007222134,
0.0074220086,
0.042346757,
0.055591777,
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0.05428593,
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0.01772221,
0.034991372,
-0.100790076,
0.051674236,
-0.041653857,
-0.01632309,
0.020253954,
0.16917382,
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0.012498822,
0.0029364903,
-0.0024001603,
-0.003870904,
-0.003444505,
0.04874274,
-0.00051009655,
0.03605737,
-0.03264618,
0.096472785,
-0.006689135,
-0.029874584,
-0.038775664,
0.0069956095,
0.025477344,
0.004603777,
-0.007082222,
0.014097819,
-0.0084746815,
-0.002713297,
0.0035511046,
-0.0439191,
0.011919186,
0.047943246,
0.08751841,
-0.008961042,
0.04946229,
0.060282167,
-0.023731772,
-0.023665147,
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-0.009234205,
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-0.055698376,
0.012079086,
0.020680355,
-0.0035910795,
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0.014457593,
0.043359455,
0.088211305,
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0.056178074,
-0.014817367,
-0.034298476,
-0.034511674,
0.006229424,
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0.023478597,
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0.025330769,
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0.02501097,
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0.036830217,
0.07712493,
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-0.032992627,
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-0.04042796,
0.06198776,
0.04040131,
0.14177768,
0.011885874,
0.0911961,
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0.021972876,
-0.03211318,
0.03288603,
0.038562465,
-0.01780216,
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-0.03197993,
-0.051620934,
0.0042406716,
0.015443641,
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0.008867768,
0.032432977,
0.028728636,
0.014804042,
-0.009474054,
-0.0060561993,
0.005369963,
-0.08464021,
0.04144066,
0.016403038,
0.019267907,
-0.016123215,
0.024358045,
0.03131368,
-0.055431876,
-0.004886933,
-0.066784754,
-0.018908134,
-0.055431876,
0.0033362394,
0.012285623,
0.038669065,
-0.009114279,
-0.050634887,
0.021599777,
0.046077747,
-0.028435487,
-0.042879757,
0.0039841663,
-0.0224659,
0.08154882,
0.0032429646,
0.002088689,
0.07200814,
0.0014490905,
0.04066781,
0.036457118,
-0.031020531,
-0.01989418,
0.04066781,
0.013871294,
-0.018894808,
0.0048969267,
-0.0009710572,
-0.0023951635,
0.06641165,
-0.022852324,
-0.079683326,
-0.020054081,
-0.033578925,
0.022119451,
-0.019041384,
0.016802788,
-0.020707004,
-0.007755133,
-0.2626618,
0.04013481,
0.06385326,
0.008961042,
-0.010786563,
0.016629564,
0.026849816,
-0.042853106,
-0.06523906,
0.00017332872,
-0.043732554,
-0.012039111,
0.08698541,
0.021773001,
-0.0457846,
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0.026383441,
-0.033712175,
0.028009089,
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0.05042169,
0.07307414,
0.21799652,
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0.026356792,
0.045438148,
0.055058777,
-0.037976164,
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0.028488787,
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0.061508063,
0.033285778,
-0.012085748,
0.05926947,
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-0.0012875253,
0.017842136,
0.07211474,
0.011166325,
0.04442545,
-0.041893706,
0.0066291727,
0.04037466,
0.0050501637,
0.056444574,
0.023185449,
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0.04082771,
-0.0038375915,
-0.0066125165,
0.015696816,
-0.028142337,
-0.03520457,
-0.002115339,
0.030700732,
-0.011133013,
0.052313834
] |
4322 | https://en.wikipedia.org/wiki/Borel%20measure | Borel measure | ary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.
Hausdorff dimension and Frostman's lemma
Given a Borel measure μ on a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ rs holds for some constant s > 0 and for every ball B(x, r) in X, then the Hausdorff dimension dimHaus(X) ≥ s. A partial converse is provided by Frostman's lemma:
Lemma: Let A be a Borel subset of Rn, and let s > 0. Then the following are equivalent:
Hs(A) > 0, where Hs denotes t | [
-0.0654623,
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0.07501109,
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0.059945215,
0.062279362,
-0.017453076,
-0.0011123683,
0.06392388,
-0.022718174,
0.11203919,
0.045038484,
0.016511459,
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0.0066410545,
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0.013321895,
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0.06371168,
0.10503674,
-0.06477266,
0.0009830617,
0.025118636,
0.019601554,
-0.030662242,
-0.040078416,
0.014097735,
0.03334121,
0.007698716,
-0.005633126,
0.056868378,
0.044110127,
-0.052730568,
-0.022055063,
-0.019296523,
0.021803083,
-0.0056729126,
-0.02131238,
0.00051183865,
0.01831512,
0.015357645,
0.020224879,
-0.020702317,
0.010695977,
0.06111229,
0.040900674,
0.010649559,
0.036073226,
0.005712699,
-0.02328845,
-0.02023814,
-0.036736336,
-0.023726104,
0.004230647,
-0.035357065,
-0.030688766,
0.028593337,
0.009721204,
-0.009727835,
0.008660227,
0.033659503,
0.052040935,
-0.010967853,
-0.024813605,
-0.038884815,
0.07734524,
-0.08105866,
-0.01610033,
-0.042173844,
-0.020755367,
0.03172322,
0.06837998,
0.025728697,
-0.08015683,
-0.042332992,
-0.025609339,
-0.004562202,
-0.010947959,
-0.048831478,
-0.3227493,
-0.058937285,
-0.023275187,
-0.07363182,
-0.004028398,
-0.02161741,
-0.0048639174,
-0.040449757,
0.02226726,
0.0076058805,
0.027691506,
-0.08243793,
-0.055117767,
-0.0012035461,
0.0062398724,
0.0076655606,
-0.028168945,
0.035648834,
-0.013335157,
0.008136369,
-0.022890585,
0.04161683,
0.016922588,
-0.006634423,
0.02226726,
-0.0031017007,
0.11288798,
-0.012333861,
0.04681562,
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0.044269275,
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0.043685738,
0.05304886,
-0.0024783765,
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0.06806169,
-0.0072875875,
-0.011279514,
0.022293784,
-0.028566811,
-0.0070090806,
0.040184513,
-0.026418334,
0.0040681846,
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-0.06397693,
0.047982696,
0.00212527,
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0.016551245,
-0.037585117,
0.019827012,
-0.013242322,
-0.016564507,
-0.083392814,
-0.036550667,
0.020251403,
-0.035807982,
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-0.007884387,
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0.057133622,
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-0.012837824,
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0.03190889,
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0.0038228335,
0.02633876,
0.04291653,
0.02885858,
0.09607149,
-0.0349592,
-0.016776703,
0.009170822,
-0.009031569,
0.01376618,
0.082331836,
0.008089951,
-0.014482339,
0.02352717,
0.021948967,
-0.037160728,
0.023871988,
0.011969149,
0.035516214,
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0.035463165,
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-0.025768485,
-0.25421014,
0.02190918,
0.0070687607,
0.020198354,
0.0015782036,
0.0043500066,
0.018288596,
-0.07957329,
-0.024322903,
-0.018023351,
0.020251403,
0.06397693,
0.058512893,
0.004688193,
-0.016789965,
-0.050847333,
0.052412275,
-0.0626507,
0.0535263,
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0.04975983,
0.060369603,
0.22641253,
-0.009946661,
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-0.0023009945,
0.038566522,
0.009369755,
-0.020463599,
-0.024362689,
-0.013792704,
0.06429522,
0.11044773,
0.0060608326,
0.07607207,
0.046179034,
0.03400432,
0.052491847,
0.013394837,
0.020224879,
0.018288596,
-0.022665126,
-0.05381807,
0.007254432,
0.037929937,
-0.008070058,
0.016511459,
-0.007042236,
-0.025224734,
-0.012121665,
0.035648834,
0.023062993,
0.024946228,
-0.011140261,
0.016232952,
0.066364124,
-0.0035940604,
-0.01125299,
-0.0095753195,
-0.03047657,
-0.026365284,
-0.05344673,
-0.067849495,
-0.025476716,
0.101429425
] |
4322 | https://en.wikipedia.org/wiki/Borel%20measure | Borel measure | he s-dimensional Hausdorff measure.
There is an (unsigned) Borel measure μ satisfying μ(A) > 0, and such that
holds for all x ∈ Rn and r > 0.
Cramér–Wold theorem
The Cramér–Wold theorem in measure theory states that a Borel probability measure on is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.
References
Further reading
Gaussian measure, a fi | [
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4322 | https://en.wikipedia.org/wiki/Borel%20measure | Borel measure | nite-dimensional Borel measure
.
Wiener's lemma related
External links
Borel measure at Encyclopedia of Mathematics
Measures (measure theory) | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | Blackadder is a series of four period British sitcoms, plus several one-off instalments, which originally aired on BBC One from 1983 to 1989. All television episodes starred Rowan Atkinson as the antihero Edmund Blackadder and Tony Robinson as Blackadder's dogsbody, Baldrick. Each series was set in a different historical period, with the two protagonists accompanied by different characters, though several reappear in one series or another, e.g., Melchett (Stephen Fry) and Lord Flashheart (Rik Mayall).
The | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | first series, The Black Adder, was written by Richard Curtis and Rowan Atkinson, while subsequent series were written by Curtis and Ben Elton. The shows were produced by John Lloyd. In 2000, the fourth series, Blackadder Goes Forth, ranked at 16 in the 100 Greatest British Television Programmes, a list created by the British Film Institute. In a 2001 poll by Channel 4, Edmund Blackadder was ranked third on their list of the 100 Greatest TV Characters. In the 2004 TV poll to find Britain's Best Sitcom, Black | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | adder was voted the second-best British sitcom of all time, topped by Only Fools and Horses. It was also ranked as the 9th-best TV show of all time by Empire magazine.
Premise
Each series is set in a different period of British history, beginning in 1485 and ending in 1917, and comprises six half-hour episodes. Blackadder follows the misfortunes of Edmund Blackadder (played by Atkinson), who in each series is a member of the same British family dynasty. It is implied in each series that the Blackadder cha | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | racter is a descendant of the previous one (the end theme lyrics of series 2 episode "Head" specify that he is the great-grandson of the previous), although it is never specified how or when any of the Blackadders (who are usually bachelors) managed to father children.
In series one, Edmund Blackadder is not particularly bright, and is much the intellectual inferior of his servant, Baldrick (played by Tony Robinson). However, in subsequent series the positions are reversed: Blackadder is clever, shrewd, sc | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | heming and manipulative while Baldrick is extremely dim. Each incarnation of Blackadder and Baldrick is also saddled with tolerating the presence of a dimwitted aristocrat. In the first two series this is Lord Percy Percy, played by Tim McInnerny. Hugh Laurie plays the role in the third and fourth series, as Prince George, Prince Regent and Lieutenant George, respectively.
The first series, made in 1983, was called The Black Adder and was set in the fictional reign of "Richard IV". The second series, Black | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | adder II (1986), was set during the reign of Elizabeth I. Blackadder the Third (1987) was set during the late 18th and early 19th centuries in the reign of George III, and Blackadder Goes Forth (1989) was set in 1917 in the trenches of the Great War.
Episodes
Series 1: The Black Adder
The Black Adder, the first series of Blackadder, was written by Richard Curtis and Rowan Atkinson and produced by John Lloyd. It originally aired on BBC1 from 15 June 1983 to 20 July 1983, and was a joint production with th | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | e Australian Seven Network.
Set in 1485 at the end of the British Middle Ages, the series is written as an alternative history in which Richard III won the Battle of Bosworth Field only to be mistaken for someone else and murdered, and is succeeded by Richard IV (Brian Blessed), one of the Princes in the Tower. The series follows the exploits of Richard IV's unfavoured second son Edmund, the Duke of Edinburgh (who calls himself "The Black Adder") in his various attempts to increase his standing with his fa | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | ther and his eventual quest to overthrow him.
Conceived while Atkinson and Curtis were working on Not the Nine O'Clock News, the series dealt comically with a number of aspects of medieval life in Britain: witchcraft, royal succession, European relations, the Crusades, and the conflict between the Church and the Crown. Along with the secret history, many historical events portrayed in the series were anachronistic (for example, Constantinople had already fallen to the Ottoman Empire in 1453, predating the | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | events in the episode by 33 years); this dramatic license would continue in the subsequent Blackadders. The filming of the series was highly ambitious, with a large cast and much location shooting. The series also featured Shakespearean dialogue, often adapted for comic effect; the end credits featured the words "Additional Dialogue by William Shakespeare".
Series 2: Blackadder II
Blackadder II is set in England during the reign of Queen Elizabeth I (1558–1603), who is portrayed by Miranda Richardson. The | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | principal character is Edmund, Lord Blackadder, the great-grandson of the original Black Adder. During the series, he regularly deals with the Queen, her obsequious Lord Chamberlain Lord Melchett (Stephen Fry) – his rival – and the Queen's demented former nanny Nursie (Patsy Byrne).
Following the BBC's request for improvements (and a severe budget reduction), several changes were made. The second series was the first to establish the familiar Blackadder character: cunning, shrewd and witty, in sharp contr | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | ast to the first series' bumbling Prince Edmund. To reduce the cost of production, it was shot with virtually no outdoor scenes (the first series was shot largely on location) and several frequently used indoor sets, such as the Queen's throne room and Blackadder's front room.
A quote from this series ranked number three in a list of the top 25 television "putdowns" of the last 40 years by the Radio Times magazine: "The eyes are open, the mouth moves, but Mr. Brain has long since departed, hasn't he, Percy | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | ?"
Series 3: Blackadder the Third
Blackadder the Third is set in the late 18th and early 19th centuries, a period known as the Regency. In the series, Edmund Blackadder Esquire is the butler to the Prince Regent, the Prince of Wales (the prince is played by Hugh Laurie as a complete fop and idiot). Despite Edmund's respected intelligence and abilities, he has no personal fortune to speak of, apart from his frequently fluctuating wage packet (as well, it seems, from stealing and selling off the Prince's so | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | cks) from the Prince: "If I'm running short of cash, all I have to do is go upstairs and ask Prince Fathead for a rise." The episode titles were puns on Jane Austen novels.
As well as Rowan Atkinson and Tony Robinson in their usual roles, this series starred Hugh Laurie as the Prince Regent, and Helen Atkinson-Wood as Mrs. Miggins. The series features Dr. Samuel Johnson (Robbie Coltrane); William Pitt the Younger (Simon Osborne); the French Revolution (with Chris Barrie, Tim McInnerny as the Scarlet Pimper | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | nel, and Nigel Planer); hammy theatrical actors (Kenneth Connor and Hugh Paddick); a squirrel-hating cross-dressing highwayman (Miranda Richardson); and a duel with the Duke of Wellington (Stephen Fry).
Series 4: Blackadder Goes Forth
This series is set in 1917, on the Western Front in the trenches of the First World War. Another "big push" is planned, and Captain Blackadder's one goal is to avoid being killed, but his schemes always land him back in the trenches. Blackadder is joined by his batman Privat | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | e S. Baldrick (Tony Robinson) and idealistic Edwardian twit Lieutenant George (Hugh Laurie). General Melchett (Stephen Fry) rallies his troops from a French château from the front, where he is aided and abetted by his assistant, Captain Kevin Darling (Tim McInnerny), pencil-pusher supreme and Blackadder's nemesis, whose name is played on for maximum comedic value.
The series' tone is somewhat darker than the other Blackadders; it details the deprivations of trench warfare as well as the incompetence and l | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | ife-wasting strategies of the top brass. For example, Baldrick is reduced to making coffee from mud and cooking rats, while General Melchett hatches a plan for the troops to walk very slowly toward the German lines, because "it'll be the last thing Fritz will expect."
The final episode, "Goodbyeee", is known for being extraordinarily poignant for a comedy – especially the final scene, which sees the main characters (Blackadder, Baldrick, George, and Darling) finally going "over the top" and charging off in | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | to the fog and smoke of no man's land presumably to die. In a list of the 100 Greatest British Television Programmes, drawn up by the British Film Institute in 2000 and voted for by industry professionals, Blackadder Goes Forth was placed 16th.
Specials
Pilot episode
The Blackadder pilot was shot but never broadcast on terrestrial TV in the UK (although some scenes were shown in the 25th anniversary special Blackadder Rides Again). One notable difference in the pilot, as in many pilots, is the casting. B | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | aldrick is played not by Tony Robinson, but by Philip Fox. Another significant difference is that the character of Prince Edmund presented in the pilot is much closer to the intelligent, conniving Blackadder of the later series than the snivelling, weak buffoon of the original. Set in the year 1582, the script of the pilot is roughly the same as the episode "Born to Be King", albeit with some different jokes, with some lines appearing in other episodes of the series.
Blackadder: The Cavalier Years
This sp | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | ecial, set in the English Civil War, was shown as part of Comic Relief's Red Nose Day on Friday 5 February 1988. The 15-minute episode is set in November 1648, during the last days of the Civil War. Sir Edmund Blackadder and his servant, Baldrick, are the last two men loyal to the defeated King Charles I of England (played by Stephen Fry, portrayed as a soft-spoken, ineffective, naive character, with the voice and mannerisms of Charles I's namesake, the current Prince of Wales). However, due to a misunderst | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | anding between Oliver Cromwell (guest-star Warren Clarke) and Baldrick, the king is arrested and sent to the Tower of London. The rest of the episode revolves around Blackadder's attempts to save the king, as well as improve his standing.
Blackadder's Christmas Carol
The second special was broadcast on Friday 23 December 1988. In a twist on Charles Dickens' A Christmas Carol, Ebenezer Blackadder is the "kindest and loveliest" man in England. The Spirit of Christmas shows Blackadder the contrary antics of | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | his ancestors and descendants, and reluctantly informs him that if he turns evil his descendants will enjoy power and fortune, while if he remains the same a future Blackadder will live shamefully subjugated to a future incompetent Baldrick. This remarkable encounter causes him to proclaim, "Bad guys have all the fun", and adopt the personality with which viewers are more familiar.
Blackadder: Back & Forth
Blackadder: Back & Forth was originally shown in the Millennium Dome in 2000, followed by a screenin | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | g on Sky One in the same year (and later on BBC1). It is set on the turn of the millennium, and features Lord Blackadder placing a bet with his friends – modern versions of Queenie (Miranda Richardson), Melchett (Stephen Fry), George (Hugh Laurie) and Darling (Tim McInnerny) – that he has built a working time machine. While this is intended as a clever con trick, the machine surprisingly works, sending Blackadder and Baldrick back to the Cretaceous period, where they manage to cause the extinction of the di | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | nosaurs through the use of Baldrick's best-worst-and-only pair of underpants as a weapon against a hungry T. Rex. Finding that Baldrick has forgotten to write dates on the machine's dials, the rest of the film follows their attempts to find their way back to 1999, often creating huge historical anomalies in the process that must be corrected before the end. The film includes cameo appearances from Kate Moss and Colin Firth.
The Big Night In
Broadcast as part of Children in Need and Comic Relief's joint sp | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | ecial The Big Night In during the 2020 COVID-19 pandemic, Fry resumed the role of Lord Melchett (an intellectually-brilliant version), Head of the Royal Household, under lockdown at Melchett Manor, to help Prince William deal with educating his children via Zoom and discussing Tiger King, before they both step outside to clap for the National Health Service. Melchett is said to be isolating with Lord Blackadder, both grandsons to their First World War counterparts
Live stage performances
In 1998, as part | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | of Prince Charles' 50th Birthday Gala televised on ITV, Atkinson returned to the Cavalier incarnation of Blackadder reading aloud a letter to the Privy Council of King Charles I. He colourfully refuses their invitation to stage a royal gala, calling such occasions "very, very, very dull" and asserting that there was "more musical talent on display when my servant Baldrick breaks wind."
In 2000, on the BBC's annual Royal Variety Performance, Atkinson portrayed Blackadder as a present-day officer in "Her Maj | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | esty's Royal Regiment of Shirkers" and delivered a monologue titled "Blackadder: The Army Years," proposing that Britain regain her former greatness by invading (or at least buying) France.
In 2012, as part of the Prince's Trust charity show We Are Most Amused, Atkinson and Robinson reprised their roles as Blackadder and Baldrick in a comedy sketch featuring Miranda Hart as leader of a government inquiry into the recent banking crisis. Blackadder, chief executive of a fictional British bank, appearing with | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | Baldrick as his gardener, convinces the panel to publicly blame the entire crisis on Baldrick, to the latter's consternation.
Chronological order
Production
Series development
Rowan Atkinson and Richard Curtis developed the idea for the sitcom while working on Not the Nine O'Clock News. Eager to avoid comparisons to the critically acclaimed Fawlty Towers, they proposed the idea of a historical sitcom. An unaired pilot episode was made in 1982, and a six-episode series was commissioned. The budget for th | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | e series was considerable, with much location shooting particularly at Alnwick Castle in Northumberland and the surrounding countryside in February 1983. The series also used large casts of extras, horses and expensive medieval-style costumes. Atkinson has said about the making of the first series:
The first series was odd, it was very extravagant. It cost a million pounds for the six programmes ... [which] was a lot of money to spend ... It looked great, but it wasn't as consistently funny as we would hav | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | e liked.
Due to the high cost of the first series, the then-controller of programming of BBC1, Michael Grade, was reluctant to sign off a second series without major improvements to the show and drastic cost-cutting, leaving a gap of three years between the two series.<ref name="Lewisohn">Lewisohn, Mark, [https://web.archive.org/web/20050408071319/http://www.bbc.co.uk/comedy/guide/articles/b/blackadderthe_7770760.shtml The Black Adder] at the former BBC Guide to Comedy. Retrieved 17 April 2008</ref> Atkins | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | on did not wish to continue writing for the second series.
A chance meeting between Richard Curtis and comedian Ben Elton led to the decision to collaborate on a new series of Blackadder. Recognising the main faults of the first series, Curtis and Elton agreed that Blackadder II would be a studio-only production (along with the inclusion of a live audience during recording, instead of showing the episodes to an audience after taping). Besides adding a greater comedy focus, Elton suggested a major change in | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | character emphasis: Baldrick would become the stupid sidekick, while Edmund Blackadder evolved into a cunning sycophant. This led to the familiar set-up that was maintained in the following series.
Only in the Back & Forth millennium special was the shooting once again on location, because this was a production with a budget estimated at £3 million, and was a joint venture between Tiger Aspect, Sky Television, the New Millennium Experience Company and the BBC, rather than the BBC alone.'Black to the Futur | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | e – Interview with Tony Robinson' in Skyview, January 2000
Casting
Each series tended to feature the same set of regular actors in different period settings, although throughout the four series and specials, only Blackadder and Baldrick were constant characters. Several regular cast members recurred as characters with similar names, implying, like Blackadder, that they were descendants.
Recurring cast
Various actors have appeared in more than one of the Blackadder series and/or specials. These are:
Ma | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | in cast
Rowan Atkinson as Edmund Blackadder, the series' protagonist.
Tony Robinson as S. Baldrick, his servant.
Stephen Fry as Melchett in two series, first as Lord Melchett, the sycophantic adviser to Queen Elizabeth I in series two and secondly as General Melchett, a blustering buffoon and presumed descendant in series four. Fry also appeared as Arthur Wellesley, The Duke of Wellington in series three and as various characters in Blackadder Back & Forth.
Tim McInnerny as Lord Percy Percy, Blackadder's di | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | mwitted sidekick in series one and two before a change of character to antagonistic rival Captain Kevin Darling in series four. He also appeared as The Scarlet Pimpernel (alias Lord Topper and Le Comte de Frou Frou) for one episode in the third series, and reprised his role as Darling in Blackadder: Back & Forth.
Hugh Laurie played George in series three and four, first as The Prince Regent, and later Lieutenant George in series four. Laurie also appeared twice in series two; firstly as Simon "Farters Parte | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | rs" Partridge and then as Prince Ludwig the Indestructible in the final instalment of Blackadder II. He reprised his role as George in Blackadder: Back & Forth.
Miranda Richardson was only a regular cast member for series two, in which she played Queen Elizabeth I, reprising the role in Blackadder's Christmas Carol and Back & Forth. However, she also played significant one-off roles as Amy Hardwood (a.k.a. The Shadow) in "Amy and Amiability" in the third series and Mary Fletcher-Brown, a dutiful nurse in "G | [
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4326 | https://en.wikipedia.org/wiki/Blackadder | Blackadder | eneral Hospital" from the fourth. She reappeared as Queenie and additional characters in Christmas Carol and Back and Forth.
Non-recurring cast
Brian Blessed, Elspet Gray and Robert East appeared in all six episodes of the first series as the Black Adder's father, mother and brother respectively. Gray had also appeared in the non-broadcast pilot.
Patsy Byrne played Nursie in all six episodes of Blackadder II, but never featured in either of the subsequent series, either as a regular character or one-off. S | [
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