Maximum Path Sum in a Binary Tree
Last Updated :
03 Feb, 2025
Given a binary tree, the task is to find the maximum path sum. The path may start and end at any node in the tree.
Example:
Input:
Output: 42
Explanation: Max path sum is represented using green colour nodes in the above binary tree.
Input:
Output: 31
Explanation: Max path sum is represented using green colour nodes in the above binary tree.
Approach:
If the maximum sum path in a binary tree passes through the root, there are four possible cases to consider:
- The path consists only the root itself, without involving any children.
- The path starts at the root and extends downward through its right child, possibly continuing to the bottom of the right subtree.
- The path starts at the root and extends downward through its left child, possibly continuing the bottom of the left subtree.
- The path includes the root and spans both the left and right children.
The idea is to keep track of all four paths for each subtree of the tree and pick up the max one in the end.
Implementation:
For each subtree, we want to find the maximum path sum that starts at its root and goes down through either its left or right child. To do this, we create a recursive function (lets say MPS(current root)) that calculates and returns the maximum path sum starting from the root and extending downward through one of its children.
Within the same function, we can also calculate the maximum path sum for the current subtree and compare it with the final answer. This is done by combining the MPS(current root -> left) and MPS(current root -> right) with the value of the current root.
Note that if the MPS() from the left or right child is negative, we simply ignore it while calculating the maximum path sum for the current subtree.
C++
//Driver Code Starts
// C++ program to find maximum path sum in Binary Tree
#include <iostream>
using namespace std;
class Node {
public:
int data;
Node *left, *right;
// Constructor to initialize a new node
Node(int value) {
data = value;
left = nullptr;
right = nullptr;
}
};
//Driver Code Ends
// Returns the maximum path sum in the subtree with the current node as an endpoint.
// Also updates 'res' with the maximum path sum.
int maxPathSumUtil(Node* root, int& res) {
// Base case: return 0 for a null node
if (root == NULL)
return 0;
// Calculate maximum path sums for left and right subtrees
int l = max(0, maxPathSumUtil(root->left, res));
int r = max(0, maxPathSumUtil(root->right, res));
// Update 'res' with the maximum path sum passing through the current node
res = max(res, l + r + root->data);
// Return the maximum path sum rooted at this node
return root->data + max(l, r);
}
// Returns maximum path sum in tree with given root
int maxPathSum(Node* root) {
int res = root->data;
// Compute maximum path sum and store it in 'res'
maxPathSumUtil(root, res);
return res;
}
//Driver Code Starts
int main() {
// Representation of input binary tree:
// 10
// / \
// 2 10
// / \ \
// 20 1 -25
// / \
// 3 4
Node* root = new Node(10);
root->left = new Node(2);
root->right = new Node(10);
root->left->left = new Node(20);
root->left->right = new Node(1);
root->right->right = new Node(-25);
root->right->right->left = new Node(3);
root->right->right->right = new Node(4);
cout << maxPathSum(root);
return 0;
}
//Driver Code Ends
//Driver Code Starts
// Java program to find maximum path sum in Binary Tree
class Node {
int data;
Node left, right;
// Constructor to initialize a new node
Node(int value) {
data = value;
left = null;
right = null;
}
}
//Driver Code Ends
class GfG {
// Returns the maximum path sum in the subtree with the current node as an endpoint.
// Also updates 'res' with the maximum path sum.
static int maxPathSumUtil(Node root, int[] res) {
// Base case: return 0 for a null node
if (root == null) return 0;
// Calculate maximum path sums for left and right subtrees
int l = Math.max(0, maxPathSumUtil(root.left, res));
int r = Math.max(0, maxPathSumUtil(root.right, res));
// Update 'res' with the maximum path sum passing through the current node
res[0] = Math.max(res[0], l + r + root.data);
// Return the maximum path sum rooted at this node
return root.data + Math.max(l, r);
}
// Returns maximum path sum in tree with given root
static int maxPathSum(Node root) {
int[] res = {root.data};
// Compute maximum path sum and store it in 'res'
maxPathSumUtil(root, res);
return res[0];
}
//Driver Code Starts
public static void main(String[] args) {
// Representation of input binary tree:
// 10
// / \
// 2 10
// / \ \
// 20 1 -25
// / \
// 3 4
Node root = new Node(10);
root.left = new Node(2);
root.right = new Node(10);
root.left.left = new Node(20);
root.left.right = new Node(1);
root.right.right = new Node(-25);
root.right.right.left = new Node(3);
root.right.right.right = new Node(4);
System.out.println(maxPathSum(root));
}
}
//Driver Code Ends
//Driver Code Starts
# Python program to find maximum path sum in Binary Tree
class Node:
# Constructor to initialize a new node
def __init__(self, value):
self.data = value
self.left = None
self.right = None
//Driver Code Ends
# Returns the maximum path sum in the subtree with the current node as an endpoint.
# Also updates 'res' with the maximum path sum.
def maxPathSumUtil(root, res):
# Base case: return 0 for a null node
if root is None:
return 0
# Calculate maximum path sums for left and right subtrees
l = max(0, maxPathSumUtil(root.left, res))
r = max(0, maxPathSumUtil(root.right, res))
# Update 'res' with the maximum path sum passing through the current node
res[0] = max(res[0], l + r + root.data)
# Return the maximum path sum rooted at this node
return root.data + max(l, r)
# Returns maximum path sum in tree with given root
def maxPathSum(root):
res = [root.data]
# Compute maximum path sum and store it in 'res'
maxPathSumUtil(root, res)
return res[0]
//Driver Code Starts
if __name__ == "__main__":
# Representation of input binary tree:
# 10
# / \
# 2 10
# / \ \
# 20 1 -25
# / \
# 3 4
root = Node(10)
root.left = Node(2)
root.right = Node(10)
root.left.left = Node(20)
root.left.right = Node(1)
root.right.right = Node(-25)
root.right.right.left = Node(3)
root.right.right.right = Node(4)
print(maxPathSum(root))
//Driver Code Ends
//Driver Code Starts
// C# program to find maximum path sum in Binary Tree
using System;
class Node {
public int data;
public Node left, right;
// Constructor to initialize a new node
public Node(int value) {
data = value;
left = null;
right = null;
}
}
class GfG {
//Driver Code Ends
// Returns the maximum path sum in the subtree with the current node as an endpoint.
// Also updates 'res' with the maximum path sum.
static int maxPathSumUtil(Node root, ref int res) {
// Base case: return 0 for a null node
if (root == null)
return 0;
// Calculate maximum path sums for left and right subtrees
int l = Math.Max(0, maxPathSumUtil(root.left, ref res));
int r = Math.Max(0, maxPathSumUtil(root.right, ref res));
// Update 'res' with the maximum path sum passing through the current node
res = Math.Max(res, l + r + root.data);
// Return the maximum path sum rooted at this node
return root.data + Math.Max(l, r);
}
// Returns maximum path sum in tree with given root
static int maxPathSum(Node root) {
int res = root.data;
// Compute maximum path sum and store it in 'res'
maxPathSumUtil(root, ref res);
return res;
}
//Driver Code Starts
static void Main(string[] args) {
// Representation of input binary tree:
// 10
// / \
// 2 10
// / \ \
// 20 1 -25
// / \
// 3 4
Node root = new Node(10);
root.left = new Node(2);
root.right = new Node(10);
root.left.left = new Node(20);
root.left.right = new Node(1);
root.right.right = new Node(-25);
root.right.right.left = new Node(3);
root.right.right.right = new Node(4);
Console.WriteLine(maxPathSum(root));
}
}
//Driver Code Ends
//Driver Code Starts
// JavaScript program to find maximum path sum in Binary Tree
class Node {
constructor(value) {
// Constructor to initialize a new node
this.data = value;
this.left = null;
this.right = null;
}
}
//Driver Code Ends
// Returns the maximum path sum in the subtree with the current node as an endpoint.
// Also updates 'res' with the maximum path sum.
function maxPathSumUtil(root, res) {
// Base case: return 0 for a null node
if (root === null) return 0;
// Calculate maximum path sums for left and right subtrees
const l = Math.max(0, maxPathSumUtil(root.left, res));
const r = Math.max(0, maxPathSumUtil(root.right, res));
// Update 'res' with the maximum path sum passing through the current node
res.value = Math.max(res.value, l + r + root.data);
// Return the maximum path sum rooted at this node
return root.data + Math.max(l, r);
}
// Returns maximum path sum in tree with given root
function maxPathSum(root) {
const res = { value: root.data };
// Compute maximum path sum and store it in 'res'
maxPathSumUtil(root, res);
return res.value;
}
//Driver Code Starts
// Driver Code
// Representation of input binary tree:
// 10
// / \
// 2 10
// / \ \
// 20 1 -25
// / \
// 3 4
const root = new Node(10);
root.left = new Node(2);
root.right = new Node(10);
root.left.left = new Node(20);
root.left.right = new Node(1);
root.right.right = new Node(-25);
root.right.right.left = new Node(3);
root.right.right.right = new Node(4);
console.log(maxPathSum(root));
//Driver Code Ends
Time Complexity: O(n), where n is the number of nodes in the Binary Tree.
Auxiliary Space: O(h), where h is the height of the tree.
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem