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Introduction to Tree Data Structure

  • Difficulty Level : Medium
  • Last Updated : 02 Sep, 2021

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, a list of references to nodes (the “children”).

Recursive Definition: : 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.

Basic Terminology In Tree 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.
  • 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 degree of its root. The degree of the node {3} is 3.
  • 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.
  • Depth of a node: The count of edges from the root to the node. Depth of node {14} is 3.
  • Height of a node: The number of edges on the longest path from that node to a leaf. Height of node {3} is 2.
  • Height of a tree: The height of a tree is the height of the root node i.e the count of edges from the root to the deepest node. The height of the above tree is 3.
  • 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 descendants

Few examples on Tree Data Structure: A code to demonstrate few of the above terminologies has been described below:


C++




// 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 y
void 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 node
void 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 node
void 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 nodes
void 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 node
void printDegrees(int Root, vector<vector<int> >& adj)
{
    for (int i = 1; i < adj.size(); i++) {
        cout << i << ": ";
        // Root has noo 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 code
int 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;
}

 
 

Output
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

 

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