Find the maximum path sum between two leaves of a binary tree

Last Updated : 14 Oct, 2024

Given a binary tree in which each node element contains a number. Find the maximum possible sum from one leaf node to another.

Examples:

Input:

Output: 27
Explanation: The maximum sum path may or may not go through the root. For example, in the following binary tree, the maximum sum is 27 (3 + 6 + 9 + 0 – 1 + 10).

To find the maximum path sum between two leaf nodes in a binary tree, traverse each node and recursively calculate the maximum sum from leaf to root in the left subtree of x (Find the maximum sum leaf to root path in a Binary Tree). Find the maximum sum from leaf to root in the right subtree of x. Add the above two calculated values and x->data compare the sum with the maximum value obtained so far and update the maximum value. Return the overall maximum value. The time complexity of the above solution is O(n²).

Approach:

We can find the maximum sum using single traversal of binary tree. The idea is to maintain two values in recursive calls. For every visited node x, we find the maximum root to leaf sum in left and right subtrees of x. We add the two values with x->data, and compare the sum with maximum path sum found so far.

Below is the implementation of above approach:

C++

// C++ program to find maximum path 
//sum between two leaves of a binary tree
#include <bits/stdc++.h>
using namespace std;

class Node {
public:
    int data; 
    Node *left, *right; 
    Node(int x) { 
        data = x; 
        left = right = nullptr;
    }
};

// mxPathSum function to calculate maximum path sum 
// between two leaves and maximum sum from node to leaf
// in a single traversal
int mxPathSum(Node *root, int &maxPathSum) {
  
    // Base case: If the node is null, 
  	// return 0
    if (root == nullptr) {
        return 0;
    }

    // Recursively calculate the maximum sum from 
  	// node to leaf for left and right subtrees
    int mxLeft = mxPathSum(root->left, maxPathSum);
    int mxRight = mxPathSum(root->right, maxPathSum);

    // If both left and right children exist, 
  	// update maxPathSum
    if (root->left != nullptr && root->right != nullptr) {
      
        // This is the sum of the left path, 
      	// right path, and the node's data
        int maxSumPathViaNode = mxLeft + mxRight + root->data;
        
        // Update the maximum sum path between
      	// two leaves
        maxPathSum = max(maxPathSum, maxSumPathViaNode);

        // Return the maximum sum from the current 
      	// node to a leaf
        return root->data + max(mxLeft, mxRight);
    }

    // If only one child exists, return the sum 
  	// from the node to leaf
    return root->data + (root->left ? mxLeft : mxRight);
}

// Function to return the maximum path 
// sum between any two leaves in the binary tree
int findMaxSum(Node* root) {
  
    // Edge case: If the tree is empty, 
  	// return 0
    if (root == nullptr) {
        return 0;
    }
    
    int maxPathSum = INT_MIN; 
    
    // Call the helper function to
  	// compute the result
    mxPathSum(root, maxPathSum);
    
    return maxPathSum;
}

int main() {
  
    // Construct a sample binary tree
    //
    //         1
    //       /   \
    //     -2     3
    //     / \   / \
    //    8  -1  4  -5
    
    Node* root = new Node(1);
    root->left = new Node(-2);
    root->right = new Node(3);
    root->left->left = new Node(8);
    root->left->right = new Node(-1);
    root->right->left = new Node(4);
    root->right->right = new Node(-5);

  	int result = findMaxSum(root);
    cout << result << endl;

    return 0;
}

// C++ program to find maximum path 
//sum between two leaves of  a binary tree

#include <stdio.h>
#include <stdlib.h>
#include <limits.h>

struct Node {
    int data;
    struct Node *left, *right;
};

// mxPathSum function to calculate maximum path
// sum between two leaves and maximum sum from node 
// to leaf in a single traversal
int mxPathSum(struct Node* root, int* maxPathSum) {
  
    // Base case: If the node is null, 
  	// return 0
    if (root == NULL) {
        return 0;
    }

    // Recursively calculate the maximum sum 
  	// from node to leaf for left and right subtrees
    int mxLeft = mxPathSum(root->left, maxPathSum);
    int mxRight = mxPathSum(root->right, maxPathSum);

    // If both left and right children exist, 
  	// update maxPathSum
    if (root->left != NULL && root->right != NULL) {
      
        // This is the sum of the left path,
      	// right path, and the node's data
        int maxSumPathViaNode = mxLeft + mxRight + root->data;
        
        // Update the maximum sum path between two leaves
        if (maxSumPathViaNode > *maxPathSum) {
            *maxPathSum = maxSumPathViaNode;
        }

        // Return the maximum sum from the current 
      	// node to a leaf
        return root->data + 
        (mxLeft > mxRight ? mxLeft : mxRight);
    }

    // If only one child exists, return the sum
  	// from the node to leaf
    return root->data + (root->left ? mxLeft : mxRight);
}

// Function to return the maximum path 
// sum between any two leaves in the binary tree
int findMaxSum(struct Node* root) {
  
    // Edge case: If the tree is empty,
  	// return 0
    if (root == NULL) {
        return 0;
    }

    int maxPathSum = INT_MIN;
    
    // Call the helper function to compute
  	// the result
    mxPathSum(root, &maxPathSum);
    
    return maxPathSum;
}

struct Node* createNode(int x) {
    struct Node* newNode = 
      (struct Node*)malloc(sizeof(struct Node));
    newNode->data = x;
    newNode->left = newNode->right = NULL;
    return newNode;
}

int main() {
  
    // Construct a sample binary tree
    //
    //         1
    //       /   \
    //     -2     3
    //     / \   / \
    //    8  -1  4  -5
    
    struct Node* root = createNode(1);
    root->left = createNode(-2);
    root->right = createNode(3);
    root->left->left = createNode(8);
    root->left->right = createNode(-1);
    root->right->left = createNode(4);
    root->right->right = createNode(-5);

    int result = findMaxSum(root);
    printf("%d\n", result); 

    return 0;
}

Java

// Java program to find maximum path 
// sum between two leaves of a binary tree

class Node {
    int data;
    Node left, right;

    Node(int x) {
        data = x;
        left = right = null;
    }
}

class GfG {
  
    // mxPathSum function to calculate maximum path sum 
  	// between two leaves and maximum sum from node to 
  	// leaf in a single traversal
    static int mxPathSum(Node root, int[] maxPathSum) {
      
        // Base case: If the node is null, return 0
        if (root == null) {
            return 0;
        }

        // Recursively calculate the maximum sum from 
        // node to leaf for left and right subtrees
        int mxLeft = mxPathSum(root.left, maxPathSum);
        int mxRight = mxPathSum(root.right, maxPathSum);

        // If both left and right children exist,
      	// update maxPathSum
        if (root.left != null && root.right != null) {
          
            // This is the sum of the left path, 
            // right path, and the node's data
            int maxSumPathViaNode = mxLeft + mxRight + root.data;

            // Update the maximum sum path between
          	// two leaves
            maxPathSum[0] = Math.max
              				(maxPathSum[0], maxSumPathViaNode);

            // Return the maximum sum from the current
          	// node to a leaf
            return root.data + Math.max(mxLeft, mxRight);
        }

        // If only one child exists, return the sum
      	// from the node to leaf
        return root.data + 
          (root.left != null ? mxLeft : mxRight);
    }

    // Function to return the maximum path 
    // sum between any two leaves in the binary tree
    static int findMaxSum(Node root) {
      
        // Edge case: If the tree is empty, return 0
        if (root == null) {
            return 0;
        }

        int[] maxPathSum = new int[] {Integer.MIN_VALUE};
        mxPathSum(root, maxPathSum);

        return maxPathSum[0];
    }

    public static void main(String[] args) {
      
        // Construct a sample binary tree
        //
        //         1
        //       /   \
        //     -2     3
        //     / \   / \
        //    8  -1  4  -5

        Node root = new Node(1);
        root.left = new Node(-2);
        root.right = new Node(3);
        root.left.left = new Node(8);
        root.left.right = new Node(-1);
        root.right.left = new Node(4);
        root.right.right = new Node(-5);

        int result = findMaxSum(root);
        System.out.println(result);
    }
}

Python

# Python program to find maximum path 
# sum between two leaves of a binary tree

class Node:
    def __init__(self, x):
        self.data = x
        self.left = None
        self.right = None

def mxPathSum(root):
    global maxPathSum
    
    # Base case: If the node is null, return 0
    if root is None:
        return 0

    # Recursively calculate the maximum sum from 
    # node to leaf for left and right subtrees
    mxLeft = mxPathSum(root.left)
    mxRight = mxPathSum(root.right)

    # If both left and right children exist, 
    # update maxPathSum
    if root.left is not None and root.right is not None:
      
        # This is the sum of the left path, 
        # right path, and the node's data
        maxSumPathViaNode = mxLeft + mxRight + root.data
        
        # Update the maximum sum path between
        # two leaves
        maxPathSum = max(maxPathSum, maxSumPathViaNode)

        # Return the maximum sum from the current node 
        # to a leaf
        return root.data + max(mxLeft, mxRight)

    # If only one child exists, return the sum from the
    # node to leaf
    return root.data + (mxLeft if root.left else mxRight)

def findMaxSum(root):
    global maxPathSum
    
    # Edge case: If the tree is empty,
    # return 0
    if root is None:
        return 0

    maxPathSum = float('-inf')
    mxPathSum(root)
    
    return maxPathSum

# Construct a sample binary tree
#
#         1
#       /   \
#     -2     3
#     / \   / \
#    8  -1  4  -5


root = Node(1)
root.left = Node(-2)
root.right = Node(3)
root.left.left = Node(8)
root.left.right = Node(-1)
root.right.left = Node(4)
root.right.right = Node(-5)

result = findMaxSum(root)
print( result)

// C# program to find maximum path 
// sum between two leaves of a binary tree
using System;

class Node {
    public int data;
    public Node left, right;

    public Node(int x) {
        data = x;
        left = right = null;
    }
}

class GfG {
  
    // mxPathSum function to calculate maximum path sum between two leaves
    // and maximum sum from node to leaf in a single traversal
    static int mxPathSum(Node root, ref int maxPathSum) {
      
        // Base case: If the node is null, return 0
        if (root == null) {
            return 0;
        }

        // Recursively calculate the maximum sum from 
        // node to leaf for left and right subtrees
        int mxLeft = mxPathSum(root.left, ref maxPathSum);
        int mxRight = mxPathSum(root.right, ref maxPathSum);

        // If both left and right children exist, 
      	// update maxPathSum
        if (root.left != null && root.right != null) {
          
            // This is the sum of the left path, 
            // right path, and the node's data
            int maxSumPathViaNode = mxLeft + mxRight + root.data;

            // Update the maximum sum path between 
          	// two leaves
            maxPathSum = Math.Max(maxPathSum, maxSumPathViaNode);

            // Return the maximum sum from the current 
          	// node to a leaf
            return root.data + Math.Max(mxLeft, mxRight);
        }

        // If only one child exists, return the sum from
      	// the node to leaf
        return root.data + 
          (root.left != null ? mxLeft : mxRight);
    }

    // Function to return the maximum path 
    // sum between any two leaves in the binary tree
    static int FindMaxSum(Node root) {
      
        // Edge case: If the tree is empty,
      	// return 0
        if (root == null) {
            return 0;
        }

        int maxPathSum = int.MinValue;

        // Call the helper function to 
      	// compute the result
        mxPathSum(root, ref maxPathSum);

        return maxPathSum;
    }

    static void Main(string[] args) {
      
        // Construct a sample binary tree
        //
        //         1
        //       /   \
        //     -2     3
        //     / \   / \
        //    8  -1  4  -5
    
        Node root = new Node(1);
        root.left = new Node(-2);
        root.right = new Node(3);
        root.left.left = new Node(8);
        root.left.right = new Node(-1);
        root.right.left = new Node(4);
        root.right.right = new Node(-5);

        int result = FindMaxSum(root);
        Console.WriteLine(result);
    }
}

JavaScript

// JavaScript program to find maximum path 
// sum between two leaves of a binary tree

class Node {
    constructor(x) {
        this.data = x;
        this.left = null;
        this.right = null;
    }
}

// mxPathSum function to calculate maximum path 
// sum between two leaves and maximum sum from node
// to leaf in a single traversal
function mxPathSum(root) {
    if (root === null) {
        return 0;
    }

    const mxLeft = mxPathSum(root.left);
    const mxRight = mxPathSum(root.right);

    // If both left and right children exist, 
    // update maxPathSum
    if (root.left !== null && root.right !== null) {
    
        // This is the sum of the left path, 
        // right path, and the node's data
        const maxSumPathViaNode = mxLeft + mxRight + root.data;
        maxPathSum = Math.max(maxPathSum, maxSumPathViaNode);

        // Return the maximum sum from the current
        // node to a leaf
        return root.data + Math.max(mxLeft, mxRight);
    }

    // If only one child exists, return the sum from 
    // the node to leaf
    return root.data + (root.left ? mxLeft : mxRight);
}

// Function to return the maximum path sum between any 
// two leaves in the binary tree
function findMaxSum(root) {
    maxPathSum = Number.NEGATIVE_INFINITY;

    mxPathSum(root);
    return maxPathSum;
}

// Construct a sample binary tree
//
//         1
//       /   \
//     -2     3
//     / \   / \
//    8  -1  4  -5
    

const root = new Node(1);
root.left = new Node(-2);
root.right = new Node(3);
root.left.left = new Node(8);
root.left.right = new Node(-1);
root.right.left = new Node(4);
root.right.right = new Node(-5);

const result = findMaxSum(root);
console.log(result);

Output

Time complexity: O(n) where n is the number of nodes
Auxiliary Space: O(h), where h is height of tree.

Count Possible Decodings of a given Digit Sequence

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