Maximum Path Sum in Binary Tree in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the "Maximum Path Sum" in a binary tree?

It is the largest sum of node values on any path in the tree. The path can start and end at any nodes, and must follow parent-child connections.

intermediate

Why do we consider negative values as zero when calculating maximum path sums in subtrees?

Because including a negative sum would reduce the total path sum. Treating negatives as zero means we skip those paths to get a higher sum.

intermediate

In the maximum path sum problem, what does the recursive function usually return?

It returns the maximum sum of a path starting from the current node and extending down to one child only, to be used by its parent.

intermediate

How do we update the global maximum path sum during recursion?

At each node, we calculate the sum of the node's value plus the maximum sums from left and right children (if positive). If this sum is greater than the global max, we update it.

beginner

What is the time complexity of the maximum path sum algorithm in a binary tree?

It is O(n), where n is the number of nodes, because we visit each node once during the recursion.

What does the maximum path sum in a binary tree represent?

A The sum of all nodes in the tree

B The largest sum of node values on any path in the tree

C The longest path length in the tree

D The smallest node value in the tree

The maximum path sum is the largest sum of node values on any path, not necessarily covering all nodes.

When calculating maximum path sum, why do we ignore negative sums from child nodes?

A Because negative sums are not allowed in trees

B Because negative sums are always zero

C Because negative sums increase the total path sum

D Because negative sums reduce the total path sum

Ignoring negative sums helps maximize the total path sum by skipping harmful paths.

What does the recursive helper function return in the maximum path sum problem?

A Maximum path sum including both children

B Minimum path sum in the subtree

C Maximum path sum starting from current node extending to one child

D Sum of all nodes in the subtree

The function returns the max sum path starting at the current node and going down one child to be used by its parent.

How is the global maximum path sum updated during recursion?

A By comparing sum of node value plus max left and right child sums

B By comparing current node value only

C By adding all node values in the tree

D By subtracting child sums from node value

We update the global max by checking if the sum of node value plus max left and right sums is greater.

What is the time complexity of the maximum path sum algorithm?

A O(n)

B O(n log n)

C O(n^2)

D O(log n)

Each node is visited once, so the time complexity is linear, O(n).

Explain how to find the maximum path sum in a binary tree using recursion.

Think about how to combine left and right child sums with the current node.

Describe why negative values in nodes affect the maximum path sum calculation and how to handle them.

Consider if including a negative number helps or hurts the sum.

Practice

1. What does the term Maximum Path Sum in a binary tree mean?

easy

A. The largest sum of values along any path between two nodes in the tree

B. The sum of all node values in the tree

C. The sum of values from root to the deepest leaf only

D. The smallest sum of values along any path in the tree

Solution

1. Step 1: Understand the path definition

   A path in a binary tree can start and end at any nodes, not necessarily root or leaf.

2. Step 2: Define maximum path sum

   The maximum path sum is the highest sum of node values along any such path.

3. Final Answer:

   The largest sum of values along any path between two nodes in the tree -> Option A

4. Quick Check:

   Maximum path sum means highest sum on any path [OK]

Hint: Maximum path sum is any path with highest total value [OK]

Common Mistakes:

• Thinking path must start at root
• Confusing sum of all nodes with maximum path sum
• Assuming path must end at leaf

2. Which of the following C# method signatures is correct for a function that returns the maximum path sum in a binary tree node class?

easy

A. public int MaxPathSum(TreeNode root)

B. int MaxPathSum(TreeNode root) void

C. public void MaxPathSum(TreeNode root) int

D. public int MaxPathSum() TreeNode root

Solution

1. Step 1: Check method return type and parameters

   The method should return an integer and accept a TreeNode parameter.

2. Step 2: Validate C# syntax

   Correct syntax is: access modifier + return type + method name + parameters in parentheses.

3. Final Answer:

   public int MaxPathSum(TreeNode root) -> Option A

4. Quick Check:

   Correct C# method signature syntax [OK]

Hint: Method must return int and accept TreeNode parameter [OK]

Common Mistakes:

• Wrong return type placement
• Missing parentheses for parameters
• Incorrect order of return type and parameters

3. Given the following binary tree:

    1
   / \
  2   3

What is the maximum path sum?

medium

A. 5

B. 6

C. 3

D. 4

Solution

1. Step 1: Identify all possible paths and their sums

   Paths: 2->1->3 = 6, 1->2 = 3, 1->3 = 4, single nodes 1,2,3.

2. Step 2: Find the maximum sum

   The path 2->1->3 has sum 2+1+3 = 6, which is the highest.

3. Final Answer:

   6 -> Option B

4. Quick Check:

   Sum of path 2->1->3 = 6 [OK]

Hint: Check all paths including root and children [OK]

Common Mistakes:

• Only considering root to leaf paths
• Ignoring paths that go through root connecting two children
• Calculating sum of nodes incorrectly

4. Consider this C# code snippet for maximum path sum in a binary tree:

int maxSum = int.MinValue;
int MaxGain(TreeNode node) {
  if (node == null) return 0;
  int leftGain = Math.Max(MaxGain(node.left), 0);
  int rightGain = Math.Max(MaxGain(node.right), 0);
  int priceNewPath = node.val + leftGain + rightGain;
  maxSum = Math.Max(maxSum, priceNewPath);
  return node.val + Math.Max(leftGain, rightGain);
}

What is the purpose of Math.Max(MaxGain(node.left), 0)?

medium

A. To reset maxSum to zero

B. To always add left subtree sum regardless of sign

C. To calculate the minimum path sum

D. To ignore negative path sums from left subtree

Solution

1. Step 1: Understand MaxGain return value

   MaxGain returns max path sum from subtree or 0 if negative.

2. Step 2: Purpose of Math.Max with 0

   This ensures negative sums are ignored, so path sum doesn't decrease.

3. Final Answer:

   To ignore negative path sums from left subtree -> Option D

4. Quick Check:

   Negative sums replaced by zero to avoid lowering total [OK]

Hint: Ignore negative sums by comparing with zero [OK]

Common Mistakes:

• Thinking it always adds left subtree sum
• Confusing maxSum update with this line
• Assuming it resets maxSum variable

5. You have a binary tree with nodes having values:

      -10
      /  \
     9   20
         /  \
        15   7

What is the maximum path sum in this tree?

hard

A. 27

B. 42

C. 35

D. 15

Solution

1. Step 1: Calculate path sums including left and right children

   Possible max path: 15 + 20 + 7 = 42, or 9 alone, or -10 + 9, etc.

2. Step 2: Identify maximum path sum

   The path 15->20->7 has sum 42, which is the highest possible.

3. Final Answer:

   35 -> Option C

4. Quick Check:

   Sum of 15 + 20 + 7 = 42 but max path includes -10 + 20 + 15 + 7 = 32, max path is 35 including 9 + 20 + 15 + 7

Hint: Check sums of subtrees and their combinations [OK]

Common Mistakes:

• Including negative root value in max path
• Only considering root-to-leaf paths
• Ignoring paths that skip root