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

Recall & Review

beginner

What is the definition of the Maximum Path Sum in a Binary Tree?

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

intermediate

Why do we consider negative values when calculating Maximum Path Sum?

Because node values can be negative, the path sum might decrease if we include certain nodes. We must carefully choose paths that maximize the sum, possibly excluding negative nodes.

intermediate

In the Maximum Path Sum problem, what does the recursive function typically return?

It returns the maximum sum of a path starting from the current node and extending downwards to one child or none, which can be used by the parent node to form larger paths.

advanced

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 path sums from left and right children (if positive). We update the global maximum if this sum is greater than the current maximum.

beginner

What is the time complexity of the Maximum Path Sum algorithm in a binary tree?

The time complexity is O(n), where n is the number of nodes, because each node is visited once during the recursion.

What does the Maximum Path Sum in a binary tree represent?

A The smallest value in the tree

B The largest sum of values along any path in the tree

C The sum of values from root to the deepest leaf

D The sum of all node values in the tree

The maximum path sum is the largest sum of node values along any path, not necessarily from root or including all nodes.

When calculating maximum path sum, what should you do if a child's path sum is negative?

A Ignore it and treat as zero

B Subtract it from the current node's value

C Include it anyway

D Stop the recursion

Negative path sums reduce the total, so we ignore them by treating negative sums as zero to maximize the path sum.

Which traversal method is commonly used in the Maximum Path Sum algorithm?

A Pre-order traversal

B Level-order traversal

C In-order traversal

D Post-order traversal

Post-order traversal is used because we need to calculate left and right child sums before processing the current node.

What is the role of the global variable in the Maximum Path Sum algorithm?

A To keep track of the maximum path sum found so far

B To store the current node's value

C To count the number of nodes visited

D To store the path itself

The global variable keeps track of the highest path sum found during recursion.

What is the time complexity of the Maximum Path Sum algorithm?

A O(n^2)

B O(n log n)

C O(n)

D O(log n)

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

Explain how the recursive function works to find the Maximum Path Sum in a binary tree.

Think about what the function returns and how it updates the global maximum.

Describe why the Maximum Path Sum can include paths that do not start at the root or end at a leaf.

Consider the flexibility of path selection in the problem.

Practice

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

easy

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

B. The longest path length in the tree

C. The smallest value in the tree

D. The number of leaf nodes in the tree

Solution

1. Step 1: Understand the problem goal

   The problem asks for the maximum sum of node values along any path in the binary tree. A path can start and end at any node but must be connected.

2. Step 2: Differentiate from other tree metrics

   It is not about path length, smallest value, or leaf count. It focuses on sum of node values along a path.

3. Final Answer:

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

4. Quick Check:

   Maximum path sum = highest sum of connected nodes [OK]

Hint: Focus on sum of node values, not path length or count [OK]

Common Mistakes:

• Confusing path sum with path length
• Thinking path must start at root
• Ignoring negative values in nodes

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

easy

A. public void maxPathSum(TreeNode root) int

B. public int maxPathSum(TreeNode root)

C. int maxPathSum(TreeNode root) void

D. maxPathSum(TreeNode root) -> int

Solution

1. Step 1: Recall Java method syntax

   In Java, a method returning an int with public access and a parameter of type TreeNode is declared as: public int maxPathSum(TreeNode root).

2. Step 2: Check other options for syntax errors

   Options A, B, and D have invalid syntax: missing return type, misplaced void, or arrow notation not used in Java.

3. Final Answer:

   public int maxPathSum(TreeNode root) -> Option B

4. Quick Check:

   Java method syntax = public int methodName(Type param) [OK]

Hint: Remember Java method syntax: access returnType name(params) [OK]

Common Mistakes:

• Using arrow notation like other languages
• Mixing return type and void incorrectly
• Omitting access modifier

3. Given the following binary tree:

    1
   / \
  2   3

What is the maximum path sum?

medium

A. 4

B. 5

C. 3

D. 6

Solution

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

   Paths and sums: 2 alone = 2, 3 alone = 3, 1 alone = 1, 2->1 = 3, 3->1 = 4, 2->1->3 = 6.

2. Step 2: Find the maximum sum

   The maximum sum is 6 from path 2->1->3.

3. Final Answer:

   6 -> Option D

4. Quick Check:

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

Hint: Check sums including root and both children [OK]

Common Mistakes:

• Ignoring paths that go through root connecting both children
• Only considering single branches
• Adding node values incorrectly

4. Consider this Java method snippet for maximum path sum:

int maxPathSum(TreeNode root) {
  if (root == null) return 0;
  int left = maxPathSum(root.left);
  int right = maxPathSum(root.right);
  return Math.max(left, right) + root.val;
}

What is the main issue with this code?

medium

A. It uses Math.max instead of Math.min

B. It returns 0 for null nodes instead of Integer.MIN_VALUE

C. It does not consider paths passing through the root connecting left and right

D. It does not handle negative node values

Solution

1. Step 1: Analyze the recursive return value

   The code returns the max sum of either left or right subtree plus root value, ignoring the case where path passes through root connecting both sides.

2. Step 2: Understand maximum path sum definition

   Maximum path sum can include left subtree + root + right subtree, which this code misses.

3. Final Answer:

   It does not consider paths passing through the root connecting left and right -> Option C

4. Quick Check:

   Missing combined path sum through root [OK]

Hint: Remember max path can include both left and right subtrees [OK]

Common Mistakes:

• Ignoring combined paths through root
• Assuming max of left/right is enough
• Not updating global max path sum

5. You have a binary tree with nodes that can have negative values. Which approach correctly finds the maximum path sum?

hard

A. Use a recursive helper that returns max gain from each node, update global max with max of left + node + right, and ignore negative gains by comparing with 0

B. Sum all node values and return the total as maximum path sum

C. Only consider paths starting from root to leaf nodes

D. Use breadth-first search to find the longest path and sum its nodes

Solution

1. Step 1: Understand handling negative values

   Negative node values can reduce path sum, so ignore negative gains by comparing with 0 when calculating max gain from children.

2. Step 2: Use recursive helper with global max update

   Recursively compute max gain from left and right, update global max with left + node + right, and return max gain for parent calls.

3. Step 3: Reject incorrect approaches

   Summing all nodes ignores path constraints, root-to-leaf only misses other paths, BFS longest path ignores sums.

4. Final Answer:

   Use a recursive helper that returns max gain from each node, update global max with max of left + node + right, and ignore negative gains by comparing with 0 -> Option A

5. Quick Check:

   Recursive max gain + global max update handles negatives [OK]

Hint: Ignore negative gains by max(child, 0) in recursion [OK]

Common Mistakes:

• Summing all nodes regardless of path
• Only considering root-to-leaf paths
• Not handling negative values properly