DP on Trees Maximum Path Sum in DSA Python - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of the DP on Trees Maximum Path Sum problem?

To find the maximum sum of values along any path in a tree, where a path can start and end at any nodes.

intermediate

In DP on Trees Maximum Path Sum, why do we use a post-order traversal?

Because we need to calculate the maximum path sums of child nodes before computing the parent's maximum path sum, ensuring all child information is ready.

intermediate

What does the 'single path max sum' represent in the DP on Trees Maximum Path Sum solution?

It represents the maximum sum of a path starting from the current node and extending down to one of its children, or just the node itself if children reduce the sum.

advanced

How do we update the global maximum path sum during the DP on Trees Maximum Path Sum computation?

At each node, we consider the sum of the node's value plus the maximum path sums from both left and right children (if any), and update the global max if this sum is larger.

intermediate

Why do we ignore negative path sums from children in the DP on Trees Maximum Path Sum problem?

Because including negative sums would reduce the total path sum, so we treat negative sums as zero to maximize the path sum.

What traversal is best suited for solving the DP on Trees Maximum Path Sum problem?

A Pre-order traversal

B Post-order traversal

C In-order traversal

D Level-order traversal

Post-order traversal processes children before the parent, which is necessary to compute maximum path sums from children first.

When calculating the maximum path sum at a node, which of these is NOT considered?

A Node's own value

B Maximum path sum from left child

C Minimum path sum from children

D Maximum path sum from right child

Minimum path sums are not used because they would reduce the total sum; we only consider maximum sums.

Why do we treat negative child path sums as zero in this problem?

A To avoid decreasing the total path sum

B Because negative sums are invalid

C To simplify the code

D Because the tree has only positive values

Negative sums reduce the total path sum, so treating them as zero helps maximize the sum.

What does the global maximum path sum represent in this problem?

A The largest sum of any path in the tree

B The sum of all nodes in the tree

C The maximum sum from root to leaf

D The minimum sum path in the tree

The global maximum path sum is the largest sum found on any path between any two nodes.

Which of these paths can be considered when finding the maximum path sum in a tree?

A Only paths from root to leaf

B Only paths that include the root

C Only paths from leaf to leaf

D Paths starting and ending at any nodes

Paths can start and end at any nodes, not necessarily including the root or leaves.

Explain how dynamic programming is applied to find the maximum path sum in a tree.

Think about how you combine child results to solve the parent's problem.

Describe the difference between the 'single path max sum' and the 'global maximum path sum' in this problem.

One is local to a node and returned up, the other is the best overall found so far.

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