Why Maximum Path Sum in Binary Tree in DSA Typescript?

The Big Idea

What if you could find the best path in a complex tree without checking every single route?

The Scenario

Imagine you have a family tree drawn on paper, and you want to find the path that gives the highest total happiness score by adding up values from each family member along the path.

Doing this by checking every possible path manually is like trying to find the best route through a maze without a map.

The Problem

Manually checking every path in a tree is slow and confusing because the number of paths grows very fast as the tree gets bigger.

You might miss some paths or add wrong numbers, making it error-prone and frustrating.

The Solution

The Maximum Path Sum algorithm smartly explores the tree using a simple rule: it calculates the best path sum from each node by combining the best sums from its children.

This way, it finds the highest total path sum without checking every path one by one.

Before vs After

✗ Before

function maxPathSumManual(root) {
  // Try all paths manually - very complex and slow
  // No clear way to do this simply
}

✓ After

function maxPathSum(root) {
  let maxSum = -Infinity;
  function helper(node) {
    if (!node) return 0;
    const left = Math.max(helper(node.left), 0);
    const right = Math.max(helper(node.right), 0);
    maxSum = Math.max(maxSum, node.val + left + right);
    return node.val + Math.max(left, right);
  }
  helper(root);
  return maxSum;
}

What It Enables

This concept allows you to quickly find the highest sum path in any tree, unlocking solutions for problems in networks, decision trees, and more.

Real Life Example

In a game where each node represents a choice with a score, this helps find the best sequence of choices to maximize your points.

Key Takeaways

Manual checking of all paths is slow and error-prone.

The algorithm uses recursion to find max sums efficiently.

It enables solving complex tree path problems quickly.

Practice

1. What does the "Maximum Path Sum in Binary Tree" problem ask you to find?

easy

A. The highest sum of node values along any path in the tree

B. The longest path length in the tree

C. The sum of all node values in the tree

D. The minimum value node in the tree

Solution

1. Step 1: Understand the problem goal

   The problem asks for the maximum sum of values along any path in the binary tree, where a path can start and end at any nodes.

2. Step 2: Differentiate from other tree sums

   This is different from longest path or total sum; it focuses on the maximum sum achievable on any path.

3. Final Answer:

   The highest sum of node values along any path in the tree -> Option A

4. Quick Check:

   Maximum path sum = highest node value sum [OK]

Hint: Focus on sums along paths, not lengths or totals [OK]

Common Mistakes:

• Confusing path sum with longest path length
• Summing all nodes instead of a path
• Looking for minimum instead of maximum

2. Which TypeScript function signature correctly represents a function to find maximum path sum in a binary tree node?

easy

A. function maxPathSum(root: number[]): number

B. function maxPathSum(root: TreeNode | null): number

C. function maxPathSum(root: string): string

D. function maxPathSum(root: TreeNode): void

Solution

1. Step 1: Identify correct parameter type

   The function should accept a binary tree node or null, so TreeNode | null is correct.

2. Step 2: Identify correct return type

   The function returns a number representing the maximum path sum, so return type is number.

3. Final Answer:

   function maxPathSum(root: TreeNode | null): number -> Option B

4. Quick Check:

   Input: TreeNode or null, Output: number [OK]

Hint: Function input is tree node or null, output is number [OK]

Common Mistakes:

• Using array or string instead of TreeNode
• Returning void instead of number
• Not allowing null as input

3. Given the following TypeScript code, what is the output?

class TreeNode {
  val: number;
  left: TreeNode | null;
  right: TreeNode | null;
  constructor(val: number, left: TreeNode | null = null, right: TreeNode | null = null) {
    this.val = val;
    this.left = left;
    this.right = right;
  }
}

function maxPathSum(root: TreeNode | null): number {
  let maxSum = -Infinity;
  function helper(node: TreeNode | null): number {
    if (!node) return 0;
    const left = Math.max(helper(node.left), 0);
    const right = Math.max(helper(node.right), 0);
    maxSum = Math.max(maxSum, node.val + left + right);
    return node.val + Math.max(left, right);
  }
  helper(root);
  return maxSum;
}

const tree = new TreeNode(1, new TreeNode(2), new TreeNode(3));
console.log(maxPathSum(tree));

medium

A. 3

B. 5

C. 1

D. 6

Solution

1. Step 1: Calculate max path sums for each node

   Node 2 and 3 have no children, so max path sums are 2 and 3. For root 1, left=2, right=3, total=1+2+3=6.

2. Step 2: Determine maximum path sum

   Maximum path sum is max of all nodes' sums: max(2,3,6) = 6.

3. Final Answer:

   6 -> Option D

4. Quick Check:

   Sum(1+2+3) = 6 [OK]

Hint: Sum node + max left + max right, ignore negatives [OK]

Common Mistakes:

• Ignoring zero floor for negative sums
• Returning sum of only root and one child
• Confusing return value with maxSum variable

4. Identify the bug in this TypeScript function for maximum path sum:

function maxPathSum(root: TreeNode | null): number {
  let maxSum = 0;
  function helper(node: TreeNode | null): number {
    if (!node) return 0;
    const left = helper(node.left);
    const right = helper(node.right);
    maxSum = Math.max(maxSum, node.val + left + right);
    return node.val + Math.max(left, right);
  }
  helper(root);
  return maxSum;
}

medium

A. Returns node.val + min(left, right) instead of max

B. helper function missing base case

C. maxSum initialized to 0 instead of -Infinity

D. Does not call helper on root

Solution

1. Step 1: Check maxSum initialization

   maxSum initialized to 0 can cause wrong result if all node values are negative.

2. Step 2: Correct initialization for negative values

   Initialize maxSum to -Infinity to handle negative node values correctly.

3. Final Answer:

   maxSum initialized to 0 instead of -Infinity -> Option C

4. Quick Check:

   maxSum must start at -Infinity for negative nodes [OK]

Hint: Initialize maxSum to -Infinity to handle negatives [OK]

Common Mistakes:

• Starting maxSum at 0 causes wrong max for negative trees
• Forgetting base case returns 0
• Using min instead of max for return

5. Given a binary tree with nodes: root= -10, left=9, right=20, right.left=15, right.right=7, what is the maximum path sum?

hard

A. 42

B. 35

C. 27

D. 20

Solution

1. Step 1: Calculate max path sums for subtrees

   Right subtree max path: 15 + 20 + 7 = 42. Left subtree max is 9. Root is -10.

2. Step 2: Determine overall max path sum

   Max path sum is max of 42 (right subtree path), 9 (left), and paths including root. The max is 42.

3. Final Answer:

   42 -> Option A

4. Quick Check:

   Sum(15+20+7) = 42 [OK]

Hint: Check all subtree sums including root, pick max [OK]

Common Mistakes:

• Ignoring path through both children and root
• Choosing sum of root and one child only
• Not considering negative root value impact