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LESSON 18

Lesson 18 - Binary Trees Part 1

The binary tree is a fundamental data structure used in computer science. The binary tree is a useful data structure for rapidly storing sorted data and rapidly retrieving stored data. A binary tree is composed of parent nodes, or leaves, each of which stores data and also links to up to two other child nodes (leaves) which can be visualized spatially as below the first node with one placed to the left and with one placed to the right. It is the relationship between the leaves linked to and the linking leaf, also known as the parent node, which makes the binary tree such an efficient data structure. It is the leaf on the left which has a lesser key value (ie, the value used to search for a leaf in the tree), and it is the leaf on the right which has an equal or greater key value. As a result, the leaves on the farthest left of the tree have the lowest values, whereas the leaves on the right of the tree have the greatest values. More importantly, as each leaf connects to two other leaves, it is the beginning of a new, smaller, binary tree. Due to this nature, it is possible to easily access and insert data in a binary tree using search and insert functions recursively called on successive leaves.

The typical graphical representation of a binary tree is essentially that of an upside down tree. It begins with a root node, which contains the original key value. The root node has two child nodes; each child node might have its own child nodes. Ideally, the tree would be structured so that it is a perfectly balanced tree, with each node having the same number of child nodes to its left and to its right. A perfectly balanced tree allows for the fastest average insertion of data or retrieval of data. The worst case scenario is a tree in which each node only has one child node, so it becomes as if it were a linked list in terms of speed. The typical representation of a binary tree looks like the following:

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10

/ \

6 14

/ \ / \

5 8 11 18

The node storing the 10, represented here merely as 10, is the root node, linking to the left and right child nodes, with the left node storing a lower value than the parent node, and the node on the right storing a greater value than the parent node. Notice that if one removed the root node and the right child nodes, that the node storing the value 6 would be the equivalent a new, smaller, binary tree.

The structure of a binary tree makes the insertion and search functions simple to implement using recursion. In fact, the two insertion and search functions are also both very similar. To insert data into a binary tree involves a function searching for an unused node in the proper position in the tree in which to insert the key value. The insert function is generally a recursive function that continues moving down the levels of a binary tree until there is an unused leaf in a position which follows the rules of placing nodes. The rules are that a lower value should be to the left of the node, and a greater or equal value should be to the right. Following the rules, an insert function should check each node to see if it is empty, if so, it would insert the data to be stored along with the key value (in most implementations, an empty node will simply be a NULL pointer from a parent node, so the function would also have to create the node). If the node is filled already, the insert function should check to see if the key value to be inserted is less than the key value of the current node, and if so, the insert function should be recursively called on the left child node, or if the key value to be inserted is greater than or equal to the key value of the current node the insert function should be recursively called on the right child node. The search function works along a similar fashion. It should check to see if the key value of the current node is the value to be searched. If not, it should check to see if the value to be searched for is less than the value of the node, in which case it should be recursively called on the left child node, or if it is greater than the value of the node, it should be recursively called on the right child node. Of course, it is also necessary to check to ensure that the left or right child node actually exists before calling the function on the node.

Because binary trees have log (base 2) n layers, the average search time for a binary tree is log (base 2) n. To fill an entire binary tree, sorted, takes roughly log (base 2) n * n. Lets take a look at the necessary code for a simple implementation of a binary tree. First, it is necessary to have a struct, or class, defined as a node.

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struct node

{

int key_value;

node *left;

node *right;

};

The struct has the ability to store the key_value and contains the two child nodes which define the node as part of a tree. In fact, the node itself is very similar to the node in a linked list. A basic knowledge of the code for a linked list will be very helpful in understanding the techniques of binary trees. Essentially, pointers are necessary to allow the arbitrary creation of new nodes in the tree.

It is most logical to create a binary tree class to encapsulate the workings of the tree into a single area, and also making it reusable. The class will contain functions to insert data into the tree and to search for data. Due to the use of pointers, it will be necessary to include a function to delete the tree in order to conserve memory after the program has finished.

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class btree

{

node *root;

btree();

~btree();

void destroy_tree(node *leaf);

void insert(int key, node *leaf);

node *search(int key, node *leaf);

public:

void insert(int key);

node *search(int key);

void destroy_tree();

};

The insert and search functions that are public members of the class are designed to allow the user of the class to use the class without dealing with the underlying design. The insert and search functions which will be called recursively are the ones which contain two parameters, allowing them to travel down the tree. The destroy_tree function without arguments is a front for the destroy_tree function which will recursively destroy the tree, node by node, from the bottom up.

The code for the class would look similar to the following:

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btree::btree()

{

root=NULL;

}

It is necessary to initialize root to NULL for the later functions to be able to recognize that it does not exist.

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btree::~btree()

{

destroy_tree();

}

The destroy_tree function will set off the recursive function destroy_tree shown below which will actually delete all nodes of the tree.

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void destroy_tree(node *leaf)

{

if(leaf!=NULL)

{

destroy_tree(leaf->left);

destroy_tree(leaf->right);

delete leaf;

}

}

The function destroy_tree goes to the bottom of each part of the tree, that is, searching while there is a non-null node, deletes that leaf, and then it works its way back up. The function deletes the leftmost node, then the right child node from the leftmost node's parent node, then it deletes the parent node, then works its way back to deleting the other child node of the parent of the node it just deleted, and it continues this deletion working its way up to the node of the tree upon which delete_tree was originally called. In the example tree above, the order of deletion of nodes would be 5 8 6 11 18 14 10. Note that it is necessary to delete all the child nodes to avoid wasting memory.

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void btree::insert(int key, node *leaf)

{

if(key< leaf->key_value)

{

if(leaf->left!=NULL)

insert(key, leaf->left);

else

{

leaf->left=new node;

leaf->left->key_value=key;

leaf->left->left=NULL; //Sets the left child of the child node to null

leaf->left->right=NULL; //Sets the right child of the child node to null

}

}

else if(key>=leaf->key_value)

{

if(leaf->right!=NULL)

insert(key, leaf->right);

else

{

leaf->right=new node;

leaf->right->key_value=key;

leaf->right->left=NULL; //Sets the left child of the child node to null

leaf->right->right=NULL; //Sets the right child of the child node to null

}

}

}

The case where the root node is still NULL will be taken care of by the insert function that is nonrecursive and available to non-members of the class. The insert function searches, moving down the tree of children nodes, following the prescribed rules, left for a lower value to be inserted and right for a greater value, until it finds an empty node which it creates using the 'new' keyword and initializes with the key value while setting the new node's child node pointers to NULL. After creating the new node, the insert function will no longer call itself.

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node *btree::search(int key, node *leaf)

{

if(leaf!=NULL)

{

if(key==leaf->key_value)

return leaf;

if(key<leaf->key_value)

return search(key, leaf->left);

else

return search(key, leaf->right);

}

else return NULL;

}

The search function shown above recursively moves down the tree until it either reaches a node with a key value equal to the value for which the function is searching or until the function reaches an uninitialized node, meaning that the value being searched for is not stored in the binary tree. It returns a pointer to the node to the previous instance of the function which called it, handing the pointer back up to the search function accessible outside the class.

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void btree::insert(int key)

{

if(root!=NULL)

insert(key, root);

else

{

root=new node;

root->key_value=key;

root->left=NULL;

root->right=NULL;

}

}

The public version of the insert function takes care of the case where the root has not been initialized by allocating the memory for it and setting both child nodes to NULL and setting the key_value to the value to be inserted. If the root node already exists, insert is called with the root node as the initial node of the function, and the recursive insert function takes over.

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node *btree::search(int key)

{

return search(key, root);

}

The public version of the search function is used to set off the search recursion at the root node, keeping it from being necessary for the user to have access to the root node.

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void btree::destroy_tree()

{

destroy_tree(root);

}

The public version of the destroy tree function is merely used to initialize the recursive destroy_tree function which then deletes all the nodes of the tree.

LESSON 19

Lesson 19 - Inheratence

The ability to use the object-oriented programming is an important feature of C++. Lesson 12 introduced the idea of the class; if you have not read it and do not know the basic details of classes, you should read it before continuing this tutorial. This tutorial is n Inheritance is an important feature of classes; in fact, it is integral to the idea of object oriented programming. Inheritance allows you to create a hierarchy of classes, with various classes of more specific natures inheriting the general aspects of more generalized classes. In this way, it is possible to structure a program starting with abstract ideas that are then implemented by specific classes. For example, you might have a class Animal from which class dog and cat inherent the traits that are general to all animals; at the same time, each of those classes will have attributes specific to the animal dog or cat.

Inheritance offers many useful features to programmers. The ability, for example, of a variable of a more general class to function as any of the more specific classes which inherit from it, called polymorphism, is handy. For now, we will concentrate on the basic syntax of inheritance. Polymorphism will be covered in its own tutorial.

Any class can inherit from any other class, but it is not necessarily good practice to use inheritance (put it in the bank rather than go on a vacation). Inheritance should be used when you have a more general class of objects that describes a set of objects. The features of every element of that set (of every object that is also of the more general type) should be reflected in the more general class. This class is called the base class. base classes usually contain functions that all the classes inheriting from it, known as derived classes, will need. base classes should also have all the variables that every derived class would otherwise contain.

Let us look at an example of how to structure a program with several classes. Take a program used to simulate the interaction between types of organisms, trees, birds, bears, and other creatures coinhabiting a forest. There would likely be several base classes that would then have derived classes specific to individual animal types. In fact, if you know anything about biology, you might wish to structure your classes to take advantage of the biological classification from Kingdom to species, although it would probably be overly complex. Instead, you might have base classes for the animals and the plants. If you wanted to use more base classes (a class can be both a derived of one class and a base of another), you might have classes for flying animals and land animals, and perhaps trees and scrub. Then you would want classes for specific types of animals: pigeons and vultures, bears and lions, and specific types of plants: oak and pine, grass and flower. These are unlikely to live together in the same area, but the idea is essentially there: more specific classes ought to inherit from less specific classes.

Classes, of course, share data. A derived class has access to most of the functions and variables of the base class. There are, however, ways to keep derivedren from accessing some attributes of its base class. The keywords public, protected, and private are used to control access to information within a class. It is important to remember that public, protected, and private control information both for specific instances of classes and for classes as general data types. Variables and functions designated public are both inheritable by derived classes and accessible to outside functions and code when they are elements of a specific instance of a class. Protected variables are not accessible by functions and code outside the class, but derived classes inherit these functions and variables as part of their own class. Private variables are neither accessible outside the class when it is a specific class nor are available to derived classes. Private variables are useful when you have variables that make sense in the context of large idea.

LESSON 20

Lesson 20 - Inheratence Syntax

Before beginning this lesson, you should have an understanding of the idea of inheritance. If you do not, please read lesson 19. This lesson will consist of an overview of the syntax of inheritance, the use of the keywords public, private, and protected, and then an example program following to demonstrate each. The syntax to denote one class as inheriting from another is simple. It looks like the following: class Bear : public Animal, in place of simply the keyword class and then the class name. The ": public base_class_name" is the essential syntax of inheritance; the function of this syntax is that the class will contain all public and protected variables of the base class. Do not confuse the idea of a derived class having access to data members of a base class and specific instances of the derived class possessing data. The data members - variables and functions - possessed by the derived class are specific to the type of class, not to each individual object of that type. So, two different Bear objects, while having the same member variables and functions, may have different information stored in their variables; furthermore, if there is a class Animal with an object, say object BigAnimal, of that type, and not of a more specific type inherited from that class, those two bears will not have access to the data within BigAnimal. They will simply possess variables and functions with the same name and of the same type. A quick example of inheritance:

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class Animal

{

public:

int legs;

int arms;

int age;

Animal();

~Animal();

void eat();

void sleep();

void drink();

};

//The class Animal contains information and functions

//related to all animals (at least, all animals this lesson uses)

class Cat : public Animal

{

public:

int fur_color;

void Purr();

void fish();

void Mark_territory();

};

//For those of you familiar with cats

//eat of the above operations is unique

//to your friendly furry friends

//(or enemies, as the case may be)

A discussion of the keywords public, private, and protected is useful when discussing inheritance. The three keywords are used to control access to functions and variables stored within a class.

public:

The most open level of data hiding, anything that is public is available to all derived classes of a base class, and the public variables and data for each object of both the base and derived class is accessible by code outside the class. Functions marked public are generally those the class uses to give information to and take information from the outside world; they are typically the interface with the class. The rest of the class should be hidden from the user (This hidden nature and the highly focused nature of classes is known collectively as encapsulation). The syntax for public is:

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public:

Everything following is public until the end of the class or another data hiding keyword is used.

protected:

Variables and functions marked protected are inherited by derived classes; however, these derived classes hide the data from code outside of any instance of the object. Keep in mind, even if you have another object of the same type as your first object, the second object cannot access a protected variable in the first object. Instead, the second object will have its own variable with the same name - but not necessarily the same data. Protected is a useful level of protection for important aspects to a class that must be passed on without allowing it to be accessed. The syntax is the same as that of public. specifically,

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protected:

private:

Private is the highest level of data-hiding. Not only are the functions and variables marked private not accessible by code outside the specific object in which that data appears, but private variables and functions are not inherited. The level of data protection afforded by protected is generally more flexible than that of the private level. Of course, there is a certain joy in protecting your data with the keyword private. The syntax remains the same.

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