Maximum Path Sum in Binary Tree in DSA Typescript - Time & Space Complexity

Time Complexity: Maximum Path Sum in Binary Tree

O(n)

Understanding Time Complexity

We want to understand how the time needed to find the maximum path sum in a binary tree grows as the tree gets bigger.

Specifically, how does the number of steps change when the number of nodes increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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;
}
    

This code finds the maximum sum of values along any path in a binary tree, where a path can start and end at any node.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls visiting each node once.
• How many times: Each node is visited exactly one time during the recursion.

How Execution Grows With Input

As the number of nodes grows, the function visits each node once, so the total steps grow roughly the same as the number of nodes.

Input Size (n)	Approx. Operations
10	About 10 visits
100	About 100 visits
1000	About 1000 visits

Pattern observation: The number of operations grows linearly with the number of nodes.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the maximum path sum grows directly in proportion to the number of nodes in the tree.

Common Mistake

[X] Wrong: "The recursion visits nodes multiple times, so the time is more than linear."

[OK] Correct: Each node is visited once, and results are combined on the way back, so no repeated full visits happen.

Interview Connect

Understanding this time complexity helps you explain how your solution scales and shows you can analyze recursive tree problems clearly.

Self-Check

"What if the tree was balanced vs very skewed? How would that affect the time complexity?"

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