Longest path to the bottom of a Binary Tree forming an Arithmetic Progression
Given a Binary Tree consisting of N nodes, the task is to find the length of the longest path from any node to the bottom of the tree such that all the node values form an Arithmetic Progression.
Examples:
Input:
Output: 4
Explanation:
From the above given tree, the longest path with node values forming an AP is {6, 9, 12, 15} (Length = 4).Input:
Output: 2
Approach: The given problem can be solved by using recursion and performing the DFS Traversal on the given tree. The idea is to keep track of the difference between the current root node and the next descendant node and update the length of the longest path accordingly. Follow the steps below to solve the given problem:
- Initialize a variable, say maxLength as 1 that stores the maximum length of the path from any node to the bottom of the tree forming an Arithmetic Progression.
- Define a function, say dfs(root, currentDifference, count, maxLength) that takes the current root node, current difference, count of nodes forming AP, and the resultant maximum length as the parameter and perform the following steps:
- If the root’s left node exists, then perform the following steps:
- Find the difference between the value of the root and its left node.
- If the difference is found to be currentDifference, then update the value of maxLength to the maximum of maxLength and (count + 1) and recursively call for the function dfs(root->left, currentDifference, count + 1, maxLength).
- Otherwise, recursively call for the dfs(root->left, difference, 2, maxLength).
- If the root’s right node exists, then perform the following steps:
- Find the difference between the value of the root and its right node.
- If the difference is found to be currentDifference, then update the value of maxLength to the maximum of maxLength and (count + 1) and recursively call for the function dfs(root->right, currentDifference, count + 1, maxLength).
- Otherwise, recursively call for the dfs(root->left, difference, 2, maxLength).
- If the root’s left node exists, then perform the following steps:
- If the left child of the given root node exists, then call for the dfs(root->left, difference, 2, maxLength) where the difference is the difference between the value of the root and its left node.
- If the right child of the given root node exists, then call for the dfs(root->right, difference, 2, maxLength) where the difference is the difference between the value of the root and its right node.
- After completing the above steps, print the value of maxLength as the resultant maximum length of the path from any node to the bottom of the tree forming an Arithmetic Progression.
Below is the implementation of the above approach:
C++
// C++ program for the above approach #include "bits/stdc++.h" using namespace std; // Structure of the Tree Node struct Tree { int val; Tree *left, *right; }; // Function to create a new node Tree* newNode( int data) { Tree* temp = new Tree(); temp->val = data; temp->left = temp->right = NULL; return temp; } // Function to perform DFS Traversal // to find the maximum length of a path // to a bottom node forming an AP void dfs(Tree* root, int currentDifference, int count, int & maxLength) { // If the root's left child exists if (root->left) { // Calculate the difference int difference = root->left->val - root->val; // If the difference is same // as the current difference if (difference == currentDifference) { dfs(root->left, currentDifference, count + 1, maxLength); // Update the maxLength maxLength = max(maxLength, count + 1); } // Otherwise else { dfs(root->left, difference, 2, maxLength); } } // If the root's right child exists if (root->right) { // Find the difference int difference = root->right->val - root->val; // If the difference is the same // as the current difference if (difference == currentDifference) { dfs(root->right, currentDifference, count + 1, maxLength); // Update the maxLength maxLength = max(maxLength, count + 1); } // Otherwise else { dfs(root->right, difference, 2, maxLength); } } } // Function to find the maximum length // of the path from any node to bottom // of the tree forming an AP int maximumLengthAP(Tree* root) { // Base Cases if (root == NULL) return 0; if (root->left == NULL and root->right == NULL) { return 1; } // Stores the resultant // maximum length of the path int maxLength = 2; // If the root's left child exists if (root->left) { int difference = root->left->val - root->val; dfs(root->left, difference, 2, maxLength); } // If the root's right child exists if (root->right) { int difference = root->right->val - root->val; dfs(root->right, difference, 2, maxLength); } // Return the maximum length obtained return maxLength; } // Driver Code int main() { // Given Tree Tree* root = newNode(6); root->right = newNode(9); root->right->left = newNode(7); root->right->right = newNode(12); root->right->right->right = newNode(15); cout << maximumLengthAP(root); return 0; } |
Java
// Java program for the above approach import java.lang.*; import java.util.*; class GFG{ static int maxLength; // TreeNode class static class Node { public int val; public Node left, right; }; static Node newNode( int key) { Node temp = new Node(); temp.val = key; temp.left = temp.right = null ; return temp; } // Function to perform DFS Traversal // to find the maximum length of a path // to a bottom node forming an AP static void dfs(Node root, int currentDifference, int count) { // If the root's left child exists if (root.left != null ) { // Calculate the difference int difference = root.left.val - root.val; // If the difference is same // as the current difference if (difference == currentDifference) { dfs(root.left, currentDifference, count + 1 ); // Update the maxLength maxLength = Math.max(maxLength, count + 1 ); } // Otherwise else { dfs(root.left, difference, 2 ); } } // If the root's right child exists if (root.right != null ) { // Find the difference int difference = root.right.val - root.val; // If the difference is the same // as the current difference if (difference == currentDifference) { dfs(root.right, currentDifference, count + 1 ); // Update the maxLength maxLength = Math.max(maxLength, count + 1 ); } // Otherwise else { dfs(root.right, difference, 2 ); } } } // Function to find the maximum length // of the path from any node to bottom // of the tree forming an AP static int maximumLengthAP(Node root) { // Base Cases if (root == null ) return 0 ; if (root.left == null && root.right == null ) { return 1 ; } // Stores the resultant // maximum length of the path maxLength = 2 ; // If the root's left child exists if (root.left != null ) { int difference = root.left.val - root.val; dfs(root.left, difference, 2 ); } // If the root's right child exists if (root.right != null ) { int difference = root.right.val - root.val; dfs(root.right, difference, 2 ); } // Return the maximum length obtained return maxLength; } // Driver code public static void main(String[] args) { // Given Tree Node root = newNode( 6 ); root.right = newNode( 9 ); root.right.left = newNode( 7 ); root.right.right = newNode( 12 ); root.right.right.right = newNode( 15 ); System.out.println(maximumLengthAP(root)); } } // This code is contributed by offbeat |
Python3
# Python3 program for the above approach maxLength = 2 class Node: # Constructor to set the data of # the newly created tree node def __init__( self , key): self .val = key self .left = None self .right = None # Function to perform DFS Traversal # to find the maximum length of a path # to a bottom node forming an AP def dfs(root, currentDifference, count): global maxLength # If the root's left child exists if (root.left ! = None ): # Calculate the difference difference = root.left.val - root.val # If the difference is same # as the current difference if (difference = = currentDifference): dfs(root.left, currentDifference, count + 1 ) # Update the maxLength maxLength = max (maxLength, count + 1 ) # Otherwise else : dfs(root.left, difference, 2 ) # If the root's right child exists if (root.right ! = None ): # Find the difference difference = root.right.val - root.val # If the difference is the same # as the current difference if (difference = = currentDifference): dfs(root.right, currentDifference, count + 1 ) # Update the maxLength maxLength = max (maxLength, count + 1 ) # Otherwise else : dfs(root.right, difference, 2 ) # Function to find the maximum length # of the path from any node to bottom # of the tree forming an AP def maximumLengthAP(root): global maxLength # Base Cases if (root = = None ): return 0 if (root.left = = None and root.right = = None ): return 1 # If the root's left child exists if (root.left ! = None ): difference = root.left.val - root.val dfs(root.left, difference, 2 ) # If the root's right child exists if (root.right ! = None ): difference = root.right.val - root.val dfs(root.right, difference, 2 ) # Return the maximum length obtained return maxLength # Given Tree root = Node( 6 ) root.right = Node( 9 ) root.right.left = Node( 7 ) root.right.right = Node( 12 ) root.right.right.right = Node( 15 ) print (maximumLengthAP(root)) # This code is contributed by decode2207. |
C#
// C# program for the above approach using System; class GFG{ static int maxLength; // TreeNode class class Node { public int val; public Node left, right; }; static Node newNode( int key) { Node temp = new Node(); temp.val = key; temp.left = temp.right = null ; return temp; } // Function to perform DFS Traversal // to find the maximum length of a path // to a bottom node forming an AP static void dfs(Node root, int currentDifference, int count) { // If the root's left child exists if (root.left != null ) { // Calculate the difference int difference = root.left.val - root.val; // If the difference is same // as the current difference if (difference == currentDifference) { dfs(root.left, currentDifference, count + 1); // Update the maxLength maxLength = Math.Max(maxLength, count + 1); } // Otherwise else { dfs(root.left, difference, 2); } } // If the root's right child exists if (root.right != null ) { // Find the difference int difference = root.right.val - root.val; // If the difference is the same // as the current difference if (difference == currentDifference) { dfs(root.right, currentDifference, count + 1); // Update the maxLength maxLength = Math.Max(maxLength, count + 1); } // Otherwise else { dfs(root.right, difference, 2); } } } // Function to find the maximum length // of the path from any node to bottom // of the tree forming an AP static int maximumLengthAP(Node root) { // Base Cases if (root == null ) return 0; if (root.left == null && root.right == null ) { return 1; } // Stores the resultant // maximum length of the path maxLength = 2; // If the root's left child exists if (root.left != null ) { int difference = root.left.val - root.val; dfs(root.left, difference, 2); } // If the root's right child exists if (root.right != null ) { int difference = root.right.val - root.val; dfs(root.right, difference, 2); } // Return the maximum length obtained return maxLength; } // Driver code public static void Main(String[] args) { // Given Tree Node root = newNode(6); root.right = newNode(9); root.right.left = newNode(7); root.right.right = newNode(12); root.right.right.right = newNode(15); Console.WriteLine(maximumLengthAP(root)); } } // This code is contributed by gauravrajput1 |
Javascript
<script> // JavaScript program for the above approach let maxLength; class Node { constructor(key) { this .left = null ; this .right = null ; this .val = key; } } function newNode(key) { let temp = new Node(key); return temp; } // Function to perform DFS Traversal // to find the maximum length of a path // to a bottom node forming an AP function dfs(root, currentDifference, count) { // If the root's left child exists if (root.left != null ) { // Calculate the difference let difference = root.left.val - root.val; // If the difference is same // as the current difference if (difference == currentDifference) { dfs(root.left, currentDifference, count + 1); // Update the maxLength maxLength = Math.max(maxLength, count + 1); } // Otherwise else { dfs(root.left, difference, 2); } } // If the root's right child exists if (root.right != null ) { // Find the difference let difference = root.right.val - root.val; // If the difference is the same // as the current difference if (difference == currentDifference) { dfs(root.right, currentDifference, count + 1); // Update the maxLength maxLength = Math.max(maxLength, count + 1); } // Otherwise else { dfs(root.right, difference, 2); } } } // Function to find the maximum length // of the path from any node to bottom // of the tree forming an AP function maximumLengthAP(root) { // Base Cases if (root == null ) return 0; if (root.left == null && root.right == null ) { return 1; } // Stores the resultant // maximum length of the path maxLength = 2; // If the root's left child exists if (root.left != null ) { let difference = root.left.val - root.val; dfs(root.left, difference, 2); } // If the root's right child exists if (root.right != null ) { let difference = root.right.val - root.val; dfs(root.right, difference, 2); } // Return the maximum length obtained return maxLength; } // Given Tree let root = newNode(6); root.right = newNode(9); root.right.left = newNode(7); root.right.right = newNode(12); root.right.right.right = newNode(15); document.write(maximumLengthAP(root)); </script> |
4
Time Complexity: O(N)
Auxiliary Space: O(1)
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