DP on Trees Maximum Path Sum in DSA Python - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Maximum Path Sum in a Simple Binary Tree

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 dfs(node):
        nonlocal max_sum
        if not node:
            return 0
        left = max(dfs(node.left), 0)
        right = max(dfs(node.right), 0)
        max_sum = max(max_sum, node.val + left + right)
        return node.val + max(left, right)
    dfs(root)
    return max_sum

root = TreeNode(1, TreeNode(2), TreeNode(3))
print(maxPathSum(root))

A 5

B 1

C 6

D 3

💡 Hint

Think about the path that includes both left and right children plus the root.

The maximum path sum is 2 + 1 + 3 = 6. The path includes left child, root, and right child.

❓ Predict Output

intermediate

Maximum Path Sum with Negative Values

What is the output of this code that finds the maximum path sum in a tree with negative values?

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 dfs(node):
        nonlocal max_sum
        if not node:
            return 0
        left = max(dfs(node.left), 0)
        right = max(dfs(node.right), 0)
        max_sum = max(max_sum, node.val + left + right)
        return node.val + max(left, right)
    dfs(root)
    return max_sum

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

A 42

B 35

C 27

D 20

💡 Hint

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

The maximum path sum is 15 + 20 + 7 = 42. The path does not include the root -10 because it reduces the sum.

🔧 Debug

advanced

Identify the Error in Maximum Path Sum Implementation

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

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 dfs(node):
        if not node:
            return 0
        left = dfs(node.left)
        right = dfs(node.right)
        max_sum = max(max_sum, node.val + left + right)
        return node.val + max(left, right)
    dfs(root)
    return max_sum

root = TreeNode(1, TreeNode(2), TreeNode(3))
print(maxPathSum(root))

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

B UnboundLocalError: local variable 'max_sum' referenced before assignment

C 0

D 6

💡 Hint

Check variable scope inside nested functions.

max_sum is assigned inside dfs without 'nonlocal' keyword, so Python treats it as local but it is used before assignment causing UnboundLocalError.

🧠 Conceptual

advanced

Understanding DP State in Maximum Path Sum on Trees

In the DP solution for maximum path sum in a binary tree, what does the return value of the recursive function represent?

A The total sum of all nodes in the subtree rooted at the current node

B The maximum path sum including the current node and both children

C The maximum path sum found anywhere in the tree so far

D The maximum gain from the current node to any leaf node in one direction

💡 Hint

Think about what the function returns to its parent call.

The recursive function returns the maximum gain from the current node down to one child path, which can be extended by the parent.

🚀 Application

expert

Maximum Path Sum in a Tree with Multiple Branches (N-ary Tree)

Given an N-ary tree where each node can have multiple children, which approach correctly computes the maximum path sum where the path can start and end at any nodes?

A Use DFS to find max gain from each child, pick top two gains, update max path sum with node.val plus these two gains, return node.val plus max gain to parent

B Calculate max path sum for each child separately and return the maximum among them without combining

C Use BFS to find longest path and sum values along it, return the sum

D Sum all children values and add node.val, update max path sum with this sum, return sum to parent

💡 Hint

Consider how to extend the binary tree approach to multiple children.

For N-ary trees, we find max gains from all children, pick top two to form a path through current node, update max sum, and return max gain to parent.

Practice

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

easy

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

B. The sum of all node values in the tree

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

D. The longest path length in the tree

Solution

1. Step 1: Understand the path definition

   A path in a tree is any sequence of nodes connected by edges, not necessarily starting at the root.

2. Step 2: Define maximum path sum meaning

   The maximum path sum is the highest sum of node values along any such path, which can start and end anywhere.

3. Final Answer:

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

4. Quick Check:

   Maximum path sum = largest node value sum [OK]

Hint: Maximum path sum is about sums, not counts or single nodes [OK]

Common Mistakes:

• Confusing path sum with total tree sum
• Thinking path must start at root
• Considering only single nodes

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

def max_path_sum_helper(node: Optional[TreeNode]) -> int:

easy

A. def max_path_sum_helper(node: TreeNode) -> int:

B. def max_path_sum_helper(node: int) -> TreeNode:

C. def max_path_sum_helper(node: Optional[TreeNode]) -> int:

D. def max_path_sum_helper(node: TreeNode) -> None:

Solution

1. Step 1: Understand parameter type

   The node can be None (empty), so Optional[TreeNode] is correct to allow None.

2. Step 2: Understand return type

   The helper returns an integer representing max path sum from that node, so return type is int.

3. Final Answer:

   def max_path_sum_helper(node: Optional[TreeNode]) -> int: -> Option C

4. Quick Check:

   Optional node input and int output [OK]

Hint: Use Optional for nullable nodes and int return for sums [OK]

Common Mistakes:

• Not allowing None as input node
• Returning wrong type like None or TreeNode
• Using wrong parameter types

3. What is the output of the following code snippet?

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

def max_path_sum(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(max_path_sum(root))

medium

A. 35

B. 42

C. 27

D. 20

Solution

1. Step 1: Calculate max path sums for subtrees

   Left subtree max path sum is 9. Right subtree max path sum includes 20 + 15 + 7 = 42.

2. Step 2: Calculate max path sum including root

   Root value is -10, so max path sum including root is max(9, 0) + -10 + max(35, 0) = 34, but max_sum is updated to 42 from right subtree.

3. Final Answer:

   42 -> Option B

4. Quick Check:

   Max path sum = 15+20+7 = 42 [OK]

Hint: Ignore negative sums by max with 0 to maximize path [OK]

Common Mistakes:

• Adding negative paths instead of ignoring
• Confusing return value with max_sum
• Missing nonlocal max_sum update

4. Find the bug in this code for maximum path sum in a binary tree:

def max_path_sum(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 is not declared as nonlocal inside helper

B. helper returns wrong value for leaf nodes

C. Should return max_sum inside helper function

D. Missing base case for empty tree

Solution

1. Step 1: Check variable scope of max_sum

   max_sum is assigned inside helper but declared outside, so it needs 'nonlocal' to modify outer variable.

2. Step 2: Understand effect of missing nonlocal

   Without 'nonlocal', max_sum inside helper is local, so outer max_sum never updates, causing wrong result.

3. Final Answer:

   max_sum is not declared as nonlocal inside helper -> Option A

4. Quick Check:

   Use nonlocal to update outer max_sum [OK]

Hint: Declare nonlocal for outer variables modified inside nested functions [OK]

Common Mistakes:

• Forgetting nonlocal keyword
• Returning max_sum inside helper incorrectly
• Ignoring base case for None nodes

5. Given a binary tree where nodes can have negative values, how would you modify the maximum path sum algorithm to handle the case where all node values are negative?
Choose the best approach:

hard

A. Add absolute values of nodes to maximize sum

B. Ignore negative values and return 0 as max path sum

C. Only consider paths starting at the root

D. Return the maximum single node value if all sums are negative

Solution

1. Step 1: Understand negative values impact

   If all nodes are negative, the maximum path sum is the largest (least negative) single node value, since no path sums to 0 and longer paths sum to less.

2. Step 2: Modify algorithm to track max single node

   Keep track of the maximum node value separately and return it if all path sums are negative.

3. Final Answer:

   Return the maximum single node value if all sums are negative -> Option D

4. Quick Check:

   Handle all-negative by max single node value [OK]

Hint: Track max node value separately to handle all negatives [OK]

Common Mistakes:

• Returning 0 when all nodes are negative
• Adding absolute values incorrectly
• Restricting paths to root only