Maximum Path Sum in Binary Tree in DSA Python - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Maximum Path Sum Calculation

What is the output of the following code that calculates the maximum path sum in a binary tree?

DSA Python

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

def maxPathSum(root):
    max_sum = float('-inf')
    def helper(node):
        nonlocal max_sum
        if not node:
            return 0
        left = max(helper(node.left), 0)
        right = max(helper(node.right), 0)
        current = node.val + left + right
        max_sum = max(max_sum, current)
        return node.val + max(left, right)
    helper(root)
    return max_sum

# Tree structure:
#       -10
#       /  \
#      9   20
#          / \
#         15  7
root = TreeNode(-10, TreeNode(9), TreeNode(20, TreeNode(15), TreeNode(7)))
print(maxPathSum(root))

A 27

B 35

C 42

D 20

💡 Hint

Consider the path that includes nodes 15, 20, and 7.

The maximum path sum is 15 + 20 + 7 = 42. The path goes through the right subtree only, ignoring the left child 9 and root -10 for the maximum sum path.

🧠 Conceptual

intermediate

Understanding Path Sum Components

In the maximum path sum problem, why do we use max(helper(node.left), 0) instead of just helper(node.left)?

A To ignore negative path sums that would reduce the total sum

B To ensure the path always includes the left child

C To count zero as a valid node value

D To avoid recursion on the left child

💡 Hint

Think about how negative values affect sums.

Using max(helper(node.left), 0) means if the left path sum is negative, we treat it as zero to avoid decreasing the total path sum.

🔧 Debug

advanced

Identify the Error in Maximum Path Sum Code

What error will the following code produce when calculating maximum path sum?

DSA Python

def maxPathSum(root):
    max_sum = float('-inf')
    def helper(node):
        if not node:
            return 0
        left = max(helper(node.left), 0)
        right = max(helper(node.right), 0)
        current = node.val + left + right
        max_sum = max(max_sum, current)
        return node.val + max(left, right)
    helper(root)
    return max_sum

A No error, returns correct maximum path sum

B TypeError: unsupported operand type(s) for +: 'int' and 'NoneType'

C AttributeError: 'NoneType' object has no attribute 'left'

D UnboundLocalError: local variable 'max_sum' referenced before assignment

💡 Hint

Check variable scope inside nested functions.

max_sum is assigned inside helper without 'nonlocal' keyword, so Python treats it as local, causing UnboundLocalError when referenced before assignment.

🚀 Application

advanced

Maximum Path Sum with Negative Values

Given the binary tree below, what is the maximum path sum? Tree: -2 / \ -1 -3 / \ -4 -5

DSA Python

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

def maxPathSum(root):
    max_sum = float('-inf')
    def helper(node):
        nonlocal max_sum
        if not node:
            return 0
        left = max(helper(node.left), 0)
        right = max(helper(node.right), 0)
        current = node.val + left + right
        max_sum = max(max_sum, current)
        return node.val + max(left, right)
    helper(root)
    return max_sum

root = TreeNode(-2, TreeNode(-1), TreeNode(-3, TreeNode(-4), TreeNode(-5)))
print(maxPathSum(root))

A -1

B -2

C -3

D -4

💡 Hint

The maximum path sum might be a single node if all are negative.

The maximum path sum is -1, which is the highest single node value in the tree. Paths including other nodes reduce the sum.

🧠 Conceptual

expert

Why Use Recursion for Maximum Path Sum?

Why is recursion the preferred method to solve the maximum path sum in a binary tree?

A Because recursion uses less memory than iteration

B Because it naturally explores all paths from bottom to top, combining results efficiently

C Because iteration cannot be used on trees

D Because recursion avoids the need for any helper functions

💡 Hint

Think about how trees are structured and how recursion fits that structure.

Recursion fits tree structures well by processing child nodes first and combining their results, making it easy to calculate path sums from bottom to top.

Practice

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

easy

A. The sum of values from root to the deepest leaf

B. The largest sum of values along any path between two nodes

C. The maximum value of a single node in the tree

D. The sum of all node values in the tree

Solution

1. Step 1: Understand the path definition

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

2. Step 2: Define maximum path sum

   It is the highest sum of node values along any such path in the tree.

3. Final Answer:

   The largest sum of values along any path between two nodes -> Option B

4. Quick Check:

   Maximum path sum = largest sum on any path [OK]

Hint: Remember path can start/end anywhere, not just root or leaf [OK]

Common Mistakes:

• Thinking path must start at root
• Confusing max node value with max path sum
• Summing all nodes instead of a path

2. Which of the following Python function signatures is correct for a recursive helper to find maximum path sum in a binary tree node root?

easy

A. def maxPathSum(root: TreeNode) -> int:

B. def maxPathSum(root: int) -> TreeNode:

C. def maxPathSum(root: TreeNode) -> TreeNode:

D. def maxPathSum(root: int) -> int:

Solution

1. Step 1: Identify input type

   The function takes a TreeNode object representing the root of the tree.

2. Step 2: Identify return type

   The function returns an integer representing the maximum path sum.

3. Final Answer:

   def maxPathSum(root: TreeNode) -> int: -> Option A

4. Quick Check:

   Input=TreeNode, Output=int [OK]

Hint: Input is tree node, output is integer sum [OK]

Common Mistakes:

• Using int as input instead of TreeNode
• Returning TreeNode instead of int
• Mixing input and output types

3. Given the following binary tree and code snippet, what is the output?

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

def maxPathSum(root):
    max_sum = float('-inf')
    def helper(node):
        nonlocal max_sum
        if not node:
            return 0
        left = max(helper(node.left), 0)
        right = max(helper(node.right), 0)
        max_sum = max(max_sum, node.val + left + right)
        return node.val + max(left, right)
    helper(root)
    return max_sum

root = TreeNode(-10, TreeNode(9), TreeNode(20, TreeNode(15), TreeNode(7)))
print(maxPathSum(root))

medium

A. 35

B. 27

C. 42

D. 20

Solution

1. Step 1: Trace left and right subtree sums

   Left subtree max path sum is 9, right subtree max path sum is 20 + 15 + 7 = 42.

2. Step 2: Calculate max path sum including root

   Root (-10) + left (9) + right (20 + 15 + 7) = -10 + 9 + 22 = 21, but max path sum is updated at right subtree as 42.

3. Step 3: Identify maximum path sum

   The maximum path sum is 42, but the path that includes root is less than 42.

4. Final Answer:

   42 -> Option C

5. Quick Check:

   Max path sum = 42 [OK]

Hint: Add max(0, left/right) to avoid negative sums [OK]

Common Mistakes:

• Including negative sums from subtrees
• Confusing return value with max_sum
• Not using nonlocal for max_sum

4. Find the error in the following code snippet for maximum path sum in a binary tree:

def maxPathSum(root):
    max_sum = float('-inf')
    def helper(node):
        if not node:
            return 0
        left = helper(node.left)
        right = helper(node.right)
        max_sum = max(max_sum, node.val + left + right)
        return node.val + max(left, right)
    helper(root)
    return max_sum

medium

A. max_sum should be declared as nonlocal inside helper

B. Return statement should be outside helper function

C. Base case should return None instead of 0

D. max_sum should be initialized to 0 instead of -inf

Solution

1. Step 1: Identify variable scope issue

   max_sum is assigned inside helper but declared outside, so Python treats it as local inside helper.

2. Step 2: Fix scope with nonlocal keyword

   Adding 'nonlocal max_sum' inside helper allows updating the outer max_sum variable.

3. Final Answer:

   max_sum should be declared as nonlocal inside helper -> Option A

4. Quick Check:

   Use nonlocal to update outer variable [OK]

Hint: Use nonlocal to modify outer variables inside nested functions [OK]

Common Mistakes:

• Forgetting nonlocal causes UnboundLocalError
• Changing base case return value incorrectly
• Initializing max_sum to 0 misses negative sums

5. You want to find the maximum path sum in a binary tree where some nodes have negative values. Which approach correctly handles negative values to find the maximum path sum?

hard

A. Only consider paths starting at root to leaf

B. Add all child sums regardless of sign to current node value

C. Replace negative node values with zero before calculation

D. Ignore negative child sums by taking max(child_sum, 0) before adding to current node

Solution

1. Step 1: Understand impact of negative values

   Negative sums reduce total path sum, so ignoring them improves max sum.

2. Step 2: Use max(child_sum, 0) to ignore negatives

   This ensures only positive contributions add to current node's path sum.

3. Step 3: Avoid altering original node values

   Replacing negatives with zero changes tree data and is incorrect.

4. Final Answer:

   Ignore negative child sums by taking max(child_sum, 0) before adding to current node -> Option D

5. Quick Check:

   Ignore negatives with max(child_sum, 0) [OK]

Hint: Use max(child_sum, 0) to skip negative paths [OK]

Common Mistakes:

• Adding negative sums reduces max path sum
• Modifying node values instead of sums
• Restricting paths to root-leaf only