Introduction to Tree Data Structure

06 Jul, 2022
A tree is non-linear and a hierarchical data structure consisting of a collection of nodes such that each node of the tree stores a value and a list of references to other nodes (the “children”).
This data structure is a specialized method to organize and store data in the computer to be used more effectively. It consists of a central node, structural nodes, and sub-nodes, which are connected via edges. We can also say that tree data structure has roots, branches, and leaves connected with one another. 
A tree consists of a root, and zero or more subtrees T1, T2, ... , Tk such that there is an edge from the root of the tree to the root of each subtree.
The data in a tree are not stored in a sequential manner i.e, they are not stored linearly. Instead, they are arranged on multiple levels or we can say it is a hierarchical structure. For this reason, the tree is considered to be a non-linear data structure.
Parent Node: The node which is a predecessor of a node is called the parent node of that node. {2} is the parent node of {6, 7}.
Child Node: The node which is the immediate successor of a node is called the child node of that node. Examples: {6, 7} are the child nodes of {2}.
Root Node: The topmost node of a tree or the node which does not have any parent node is called the root node. {1} is the root node of the tree. A non-empty tree must contain exactly one root node and exactly one path from the root to all other nodes of the tree.
Leaf Node or External Node: The nodes which do not have any child nodes are called leaf nodes. {6, 14, 8, 9, 15, 16, 4, 11, 12, 17, 18, 19} are the leaf nodes of the tree.
Ancestor of a Node: Any predecessor nodes on the path of the root to that node are called Ancestors of that node. {1, 2} are the parent nodes of the node {7}
Descendant: Any successor node on the path from the leaf node to that node. {7, 14} are the descendants of the node. {2}.
Sibling: Children of the same parent node are called siblings. {8, 9, 10} are called siblings.
Level of a node: The count of edges on the path from the root node to that node. The root node has level 0.
Internal node: A node with at least one child is called Internal Node.
Neighbour of a Node: Parent or child nodes of that node are called neighbors of that node.
Subtree: Any node of the tree along with its descendant.
Number of edges: An edge can be defined as the connection between two nodes. If a tree has N nodes then it will have (N-1) edges. There is only one path from each node to any other node of the tree.
Depth of a node: The depth of a node is defined as the length of the path from the root to that node. Each edge adds 1 unit of length to the path. So, it can also be defined as the number of edges in the path from the root of the tree to the node.
Height of a node: The height of a node can be defined as the length of the longest path from the node to a leaf node of the tree.
Height of the Tree: The height of a tree is the length of the longest path from the root of the tree to a leaf node of the tree.
Degree of a Node: The total count of subtrees attached to that node is called the degree of the node. The degree of a leaf node must be 0. The degree of a tree is the maximum degree of a node among all the nodes in the tree.
Some more properties are:
Traversing in a tree is done by depth first search and breadth first search algorithm.
It has no loop and no circuit
It has no self-loop 
Its hierarchical model.
struct Node{   int data;   struct Node *left_child;   struct Node *right_child;};
Create – create a tree in data structure.Insert − Inserts data in a tree.Search − Searches specific data  in a tree to check it is present or not.Preorder Traversal – perform Traveling a tree in a pre-order manner in data structure .In order Traversal – perform Traveling a tree in an in-order manner.Post order Traversal –perform Traveling a tree in a post-order manner.
Here,
Node A is the root node
B is the parent of D and E
D and E are the siblings
D, E, and F are the leaf nodes
A and B are the ancestors of E
Few examples on Tree Data Structure: A code to demonstrate few of the above terminologies has been described below:
C++
Java
Python3
// C++ program to demonstrate some of the above// terminologies#include <bits/stdc++.h>using namespace std;// Function to add an edge between vertices x and yvoid addEdge(int x, int y, vector<vector<int> >& adj){    adj[x].push_back(y);    adj[y].push_back(x);}// Function to print the parent of each nodevoid printParents(int node, vector<vector<int> >& adj,                  int parent){    // current node is Root, thus, has no parent    if (parent == 0)        cout << node << "->Root" << endl;    else        cout << node << "->" << parent << endl;    // Using DFS    for (auto cur : adj[node])        if (cur != parent)            printParents(cur, adj, node);}// Function to print the children of each nodevoid printChildren(int Root, vector<vector<int> >& adj){    // Queue for the BFS    queue<int> q;    // pushing the root    q.push(Root);    // visit array to keep track of nodes that have been    // visited    int vis[adj.size()] = { 0 };    // BFS    while (!q.empty()) {        int node = q.front();        q.pop();        vis[node] = 1;        cout << node << "-> ";        for (auto cur : adj[node])            if (vis[cur] == 0) {                cout << cur << " ";                q.push(cur);            }        cout << endl;    }}// Function to print the leaf nodesvoid printLeafNodes(int Root, vector<vector<int> >& adj){    // Leaf nodes have only one edge and are not the root    for (int i = 1; i < adj.size(); i++)        if (adj[i].size() == 1 && i != Root)            cout << i << " ";    cout << endl;}// Function to print the degrees of each nodevoid printDegrees(int Root, vector<vector<int> >& adj){    for (int i = 1; i < adj.size(); i++) {        cout << i << ": ";        // Root has no parent, thus, its degree is equal to        // the edges it is connected to        if (i == Root)            cout << adj[i].size() << endl;        else            cout << adj[i].size() - 1 << endl;    }}// Driver codeint main(){    // Number of nodes    int N = 7, Root = 1;    // Adjacency list to store the tree    vector<vector<int> > adj(N + 1, vector<int>());    // Creating the tree    addEdge(1, 2, adj);    addEdge(1, 3, adj);    addEdge(1, 4, adj);    addEdge(2, 5, adj);    addEdge(2, 6, adj);    addEdge(4, 7, adj);    // Printing the parents of each node    cout << "The parents of each node are:" << endl;    printParents(Root, adj, 0);     // Printing the children of each node    cout << "The children of each node are:" << endl;    printChildren(Root, adj);     // Printing the leaf nodes in the tree    cout << "The leaf nodes of the tree are:" << endl;    printLeafNodes(Root, adj);     // Printing the degrees of each node    cout << "The degrees of each node are:" << endl;    printDegrees(Root, adj);     return 0;}
// java code for above approachimport java.io.*;import java.util.*; class GFG {     // Function to print the parent of each node    public static void    printParents(int node, Vector<Vector<Integer> > adj,                 int parent)    {         // current node is Root, thus, has no parent        if (parent == 0)            System.out.println(node + "->Root");        else            System.out.println(node + "->" + parent);         // Using DFS        for (int i = 0; i < adj.get(node).size(); i++)            if (adj.get(node).get(i) != parent)                printParents(adj.get(node).get(i), adj,                             node);    }     // Function to print the children of each node    public static void    printChildren(int Root, Vector<Vector<Integer> > adj)    {         // Queue for the BFS        Queue<Integer> q = new LinkedList<>();         // pushing the root        q.add(Root);         // visit array to keep track of nodes that have been        // visited        int vis[] = new int[adj.size()];         Arrays.fill(vis, 0);         // BFS        while (q.size() != 0) {            int node = q.peek();            q.remove();            vis[node] = 1;            System.out.print(node + "-> ");             for (int i = 0; i < adj.get(node).size(); i++) {                if (vis[adj.get(node).get(i)] == 0) {                    System.out.print(adj.get(node).get(i)                                     + " ");                    q.add(adj.get(node).get(i));                }            }            System.out.println();        }    }     // Function to print the leaf nodes    public static void    printLeafNodes(int Root, Vector<Vector<Integer> > adj)    {         // Leaf nodes have only one edge and are not the        // root        for (int i = 1; i < adj.size(); i++)            if (adj.get(i).size() == 1 && i != Root)                System.out.print(i + " ");         System.out.println();    }     // Function to print the degrees of each node    public static void    printDegrees(int Root, Vector<Vector<Integer> > adj)    {        for (int i = 1; i < adj.size(); i++) {            System.out.print(i + ": ");             // Root has no parent, thus, its degree is            // equal to the edges it is connected to            if (i == Root)                System.out.println(adj.get(i).size());            else                System.out.println(adj.get(i).size() - 1);        }    }     // Driver code    public static void main(String[] args)    {         // Number of nodes        int N = 7, Root = 1;         // Adjacency list to store the tree        Vector<Vector<Integer> > adj            = new Vector<Vector<Integer> >();        for (int i = 0; i < N + 1; i++) {            adj.add(new Vector<Integer>());        }         // Creating the tree        adj.get(1).add(2);        adj.get(2).add(1);         adj.get(1).add(3);        adj.get(3).add(1);         adj.get(1).add(4);        adj.get(4).add(1);         adj.get(2).add(5);        adj.get(5).add(2);         adj.get(2).add(6);        adj.get(6).add(2);         adj.get(4).add(7);        adj.get(7).add(4);         // Printing the parents of each node        System.out.println("The parents of each node are:");        printParents(Root, adj, 0);         // Printing the children of each node        System.out.println(            "The children of each node are:");        printChildren(Root, adj);         // Printing the leaf nodes in the tree        System.out.println(            "The leaf nodes of the tree are:");        printLeafNodes(Root, adj);         // Printing the degrees of each node        System.out.println("The degrees of each node are:");        printDegrees(Root, adj);    }} // This code is contributed by rj13to.
# python program to demonstrate some of the above# terminologies # Function to add an edge between vertices x and y # Function to print the parent of each node  def printParents(node, adj, parent):     # current node is Root, thus, has no parent    if (parent == 0):        print(node, "->Root")    else:        print(node, "->", parent)     # Using DFS    for cur in adj[node]:        if (cur != parent):            printParents(cur, adj, node) # Function to print the children of each node  def printChildren(Root, adj):     # Queue for the BFS    q = []     # pushing the root    q.append(Root)     # visit array to keep track of nodes that have been    # visited    vis = [0]*len(adj)     # BFS    while (len(q) > 0):        node = q[0]        q.pop(0)        vis[node] = 1        print(node, "-> ", end=" ")         for cur in adj[node]:            if (vis[cur] == 0):                print(cur, " ", end=" ")                q.append(cur)        print("\n") # Function to print the leaf nodes  def printLeafNodes(Root, adj):     # Leaf nodes have only one edge and are not the root    for i in range(0, len(adj)):        if (len(adj[i]) == 1 and i != Root):            print(i, end=" ")    print("\n") # Function to print the degrees of each node  def printDegrees(Root, adj):     for i in range(1, len(adj)):        print(i, ": ", end=" ")         # Root has no parent, thus, its degree is equal to        # the edges it is connected to        if (i == Root):            print(len(adj[i]))        else:            print(len(adj[i])-1) # Driver code  # Number of nodesN = 7Root = 1 # Adjacency list to store the treeadj = []for i in range(0, N+1):    adj.append([]) # Creating the treeadj[1].append(2)adj[2].append(1) adj[1].append(3)adj[3].append(1) adj[1].append(4)adj[4].append(1) adj[2].append(5)adj[5].append(2) adj[2].append(6)adj[6].append(2) adj[4].append(7)adj[7].append(4) # Printing the parents of each nodeprint("The parents of each node are:")printParents(Root, adj, 0) # Printing the children of each nodeprint("The children of each node are:")printChildren(Root, adj) # Printing the leaf nodes in the treeprint("The leaf nodes of the tree are:")printLeafNodes(Root, adj) # Printing the degrees of each nodeprint("The degrees of each node are:")printDegrees(Root, adj) # This code is contributed by rj13to.
The parents of each node are:
1->Root
2->1
5->2
6->2
3->1
4->1
7->4
The children of each node are:
1-> 2 3 4 
2-> 5 6 
3-> 
4-> 7 
5-> 
6-> 
7-> 
The leaf nodes of the tree are:
3 5 6 7 
The degrees of each node are:
1: 3
2: 2
3: 0
4: 1
5: 0
6: 0
7: 0
The different types of tree data structures are as follows:
1. General tree
A general tree data structure has no restriction on the number of nodes. It means that a parent node can have any number of child nodes.  
2. Binary tree  
A node of a binary tree can have a maximum of two child nodes. In the given tree diagram, node B, D, and F are left children, while E, C, and G are the right children.  
3. Balanced tree
If the height of the left sub-tree and the right sub-tree is equal or differs at most by 1, the tree is known as a balanced tree.  
   Balanced Tree                                                                      Unbalanced Tree
4. Binary search tree
As the name implies, binary search trees are used for various searching and sorting algorithms. The examples include AVL tree and red-black tree. It is a non-linear data structure. It shows that the value of the left node is less than its parent, while the value of the right node is greater than its parent.
The applications of tree data structures are as follows:
1. Spanning trees: It is the shortest path tree used in the routers to direct the packets to the destination.  
2. Binary Search Tree: It is a type of tree data structure that helps in maintaining a sorted stream of data.  
Full Binary treeComplete Binary treeSkewed Binary treeStickily Binary treeExtended Binary tree
Full Binary tree
Complete Binary tree
Skewed Binary tree
Stickily Binary tree
Extended Binary tree
3. Storing hierarchical data: Tree data structures are used to store the hierarchical data, which means data is arranged in the form of order.  
4. Syntax tree: The syntax tree represents the structure of the program’s source code, which is used in compilers.  
5. Trie: It is a fast and efficient way for dynamic spell checking. It is also used for locating specific keys from within a set.  
6. Heap: It is also a tree data structure that can be represented in a form of an array. It is used to implement priority queues.  
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