DP on Trees Maximum Path Sum in DSA Python - Interactive Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to return the maximum path sum starting from the current node.

DSA Python

def max_gain(node):
    if not node:
        return 0
    left_gain = max(max_gain(node.left), [1])
    right_gain = max(max_gain(node.right), 0)
    return node.val + left_gain + right_gain

Drag options to blanks, or click blank then click option'

Afloat('-inf')

B0

Cnode.val

D-1

Attempts:

3 left

💡 Hint

Common Mistakes

Using negative values instead of 0 causes incorrect sums.

We use 0 as the minimum gain from a subtree to avoid decreasing the path sum.

2fill in blank

medium

Complete the code to update the global maximum path sum.

DSA Python

def max_gain(node):
    if not node:
        return 0
    left_gain = max(max_gain(node.left), 0)
    right_gain = max(max_gain(node.right), 0)
    current_max_path = node.val + left_gain + right_gain
    max_gain.max_sum = max(max_gain.max_sum, [1])
    return node.val + max(left_gain, right_gain)

Drag options to blanks, or click blank then click option'

Anode.val

Bleft_gain + right_gain

Ccurrent_max_path

Dmax_gain.max_sum

Attempts:

3 left

💡 Hint

Common Mistakes

Updating with only one subtree gain misses some paths.

We update the global max with the sum of the current node and both subtree gains.

3fill in blank

hard

Fix the error in the recursive call to avoid infinite recursion.

DSA Python

def max_gain(node):
    if not node:
        return 0
    left_gain = max(max_gain([1]), 0)
    right_gain = max(max_gain(node.right), 0)
    current_max_path = node.val + left_gain + right_gain
    max_gain.max_sum = max(max_gain.max_sum, current_max_path)
    return node.val + max(left_gain, right_gain)

Drag options to blanks, or click blank then click option'

Anode.left

Bnode

Cnode.right

DNone

Attempts:

3 left

💡 Hint

Common Mistakes

Calling max_gain(node) instead of max_gain(node.left) causes infinite recursion.

The recursive call must go to the left child, not the current node, to avoid infinite recursion.

4fill in blank

hard

Fill both blanks to complete the function that initializes and returns the maximum path sum.

DSA Python

def max_path_sum(root):
    max_gain.max_sum = [1]
    max_gain(root)
    return [2]

Drag options to blanks, or click blank then click option'

Afloat('-inf')

B0

Cmax_gain.max_sum

Droot.val

Attempts:

3 left

💡 Hint

Common Mistakes

Initializing max_sum to 0 can cause wrong results if all values are negative.

Initialize max_sum to negative infinity to handle negative values and return the updated max_sum.

5fill in blank

hard

Fill all three blanks to complete the dictionary comprehension that maps nodes to their maximum gains.

DSA Python

def max_gain_map(root):
    gain_map = {node: [1] for node in nodes if [2]
    return gain_map

nodes = [root, root.left, root.right]

max_gain_map_result = max_gain_map(root)

Drag options to blanks, or click blank then click option'

Amax_gain(node)

Bnode is not None

Cnode.val > 0

Dnode.left is not None

Attempts:

3 left

💡 Hint

Common Mistakes

Including None nodes causes errors during max_gain calls.

We map each non-null node to its max gain. The condition ensures nodes exist. The third blank is a distractor here and unused.

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