DP on Trees Maximum Path Sum in DSA Go - 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.

beginner

In DP on Trees Maximum Path Sum, what does the 'path' mean?

A sequence of connected nodes in the tree, where each node is connected by an edge, and the path cannot revisit nodes.

intermediate

Why do we use recursion with memoization in DP on Trees Maximum Path Sum?

Because we calculate the maximum path sum for each subtree once and reuse it to avoid repeated calculations, improving efficiency.

intermediate

What is the significance of returning the maximum single path sum from a node to its parent in the DP solution?

It helps to build the maximum path sum that can be extended upwards to the parent, ensuring the path remains valid without cycles.

intermediate

How do we update the global maximum path sum during the DFS traversal?

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

In DP on Trees Maximum Path Sum, what does the returned value from a node's DFS call represent?

A Maximum path sum starting from this node and going down to one child

B Maximum path sum anywhere in the subtree rooted at this node

C Sum of all nodes in the subtree

D Number of nodes in the subtree

The DFS returns the maximum sum of a path starting at the current node and extending down to one child, which can be used by the parent.

Why do we consider max(0, child_path_sum) when calculating the maximum path sum at a node?

A To ignore negative path sums that reduce the total

B To include all child paths regardless of value

C To count the number of children

D To find the minimum path sum

Negative sums would reduce the total path sum, so we treat them as zero to avoid decreasing the maximum.

What is the time complexity of the DP on Trees Maximum Path Sum algorithm?

A O(n^2)

B O(n), where n is the number of nodes

C O(log n)

D O(1)

Each node is visited once during DFS, so the time complexity is linear in the number of nodes.

Which traversal method is typically used in DP on Trees Maximum Path Sum?

A Level Order Traversal

B Breadth-First Search (BFS)

C Depth-First Search (DFS)

D Inorder Traversal

DFS is used to explore each subtree fully before returning values to the parent.

Can the maximum path sum include just one node?

A No, path must include root

B No, path must have at least two nodes

C Yes, but only if the node is a leaf

D Yes, if all other paths have negative sums

A single node can be the maximum path if all other paths have smaller sums.

Explain how the DP on Trees Maximum Path Sum algorithm calculates and updates the maximum path sum during traversal.

Think about how each node combines child results and updates the answer.

Describe why we return the maximum single path sum from a node to its parent instead of the maximum path sum in the subtree.

Consider how paths connect upwards in the tree.

Practice

1. What does the "Maximum Path Sum" in a tree represent in the context of dynamic programming on trees?

easy

A. The largest sum of node values on any path between two nodes in the tree

B. The number of nodes in the longest path of the tree

C. The sum of all node values in the tree

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

Solution

1. Step 1: Understand the problem definition

   The maximum path sum is about finding the path between any two nodes that gives the highest sum of node values.

2. Step 2: Differentiate from other sums

   It is not just the sum of all nodes or the longest path length, but the maximum sum along any path.

3. Final Answer:

   The largest sum of node values on any path between two nodes in the tree -> Option A

4. Quick Check:

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

Hint: Maximum path sum means highest sum between any two nodes [OK]

Common Mistakes:

• Confusing maximum path sum with longest path length
• Thinking it is sum of all nodes
• Considering only single node values

2. Which of the following Go function signatures is correct for a recursive function that returns the maximum path sum starting from a given tree node?

easy

A. func maxPathSum(node int) int

B. func maxPathSum(node TreeNode) string

C. func maxPathSum(node *TreeNode) (int, error)

D. func maxPathSum(node *TreeNode) int

Solution

1. Step 1: Identify correct parameter type

   The function should accept a pointer to TreeNode to handle nil nodes and avoid copying.

2. Step 2: Identify correct return type

   The function returns an integer representing the maximum path sum, so return type int is correct.

3. Final Answer:

   func maxPathSum(node *TreeNode) int -> Option D

4. Quick Check:

   Pointer input and int output for max sum [OK]

Hint: Use pointer receiver and int return for max path sum [OK]

Common Mistakes:

• Passing node by value instead of pointer
• Returning string or error instead of int
• Using wrong parameter type like int

3. Given the following Go code snippet for a binary tree node and a function to compute maximum path sum, what is the output printed?

package main

import "fmt"

type TreeNode struct {
    Val int
    Left, Right *TreeNode
}

func maxPathSumUtil(node *TreeNode, maxSum *int) int {
    if node == nil {
        return 0
    }
    left := max(0, maxPathSumUtil(node.Left, maxSum))
    right := max(0, maxPathSumUtil(node.Right, maxSum))
    *maxSum = max(*maxSum, left + right + node.Val)
    return max(left, right) + node.Val
}

func maxPathSum(root *TreeNode) int {
    maxSum := -1000000
    maxPathSumUtil(root, &maxSum)
    return maxSum
}

func max(a, b int) int {
    if a > b {
        return a
    }
    return b
}

func main() {
    root := &TreeNode{Val: -10}
    root.Left = &TreeNode{Val: 9}
    root.Right = &TreeNode{Val: 20}
    root.Right.Left = &TreeNode{Val: 15}
    root.Right.Right = &TreeNode{Val: 7}
    fmt.Println(maxPathSum(root))
}

medium

A. 27

B. 35

C. 42

D. 20

Solution

1. Step 1: Calculate max path sums for subtrees

   Left subtree: 9 (updates maxSum to 9). Right subtree: 15 (maxSum=15), 7 (no change), at 20: left=15, right=7, *maxSum = max(15, 15+7+20)=42, returns max(15,7)+20=35.

2. Step 2: Compare and update maxSum

   At root (-10): left=max(0,9)=9, right=max(0,35)=35, *maxSum = max(42, 9+35+(-10))=max(42,34)=42.

3. Final Answer:

   42 -> Option C

4. Quick Check:

   Max path sum = 42 [OK]

Hint: Add max(0, child sums) and update global max [OK]

Common Mistakes:

• Not ignoring negative sums by using max(0, ...)
• Adding node values incorrectly
• Confusing max path sum with max root-to-leaf sum

4. The following Go function is intended to compute the maximum path sum in a binary tree. What is the bug causing incorrect results?

func maxPathSumUtil(node *TreeNode, maxSum *int) int {
    if node == nil {
        return 0
    }
    left := maxPathSumUtil(node.Left, maxSum)
    right := maxPathSumUtil(node.Right, maxSum)
    *maxSum = max(*maxSum, left + right + node.Val)
    return max(left, right) + node.Val
}

medium

A. It does not ignore negative sums from children before adding

B. It returns the sum of both children instead of max child sum

C. It does not update maxSum correctly

D. It does not handle nil nodes

Solution

1. Step 1: Check handling of negative child sums

   The function uses left and right sums directly without ignoring negative values, which can reduce the path sum.

2. Step 2: Identify fix by ignoring negatives

   To fix, use max(0, left) and max(0, right) to ignore negative sums before adding to node.Val.

3. Final Answer:

   It does not ignore negative sums from children before adding -> Option A

4. Quick Check:

   Ignore negative child sums for correct max path [OK]

Hint: Always ignore negative child sums with max(0, child) [OK]

Common Mistakes:

• Not using max(0, child) to ignore negatives
• Returning sum of both children instead of max child
• Not updating maxSum properly

5. You are given a binary tree where some nodes have negative values. How would you modify the standard DP on trees maximum path sum algorithm to correctly handle this case and find the maximum path sum?

hard

A. Ignore all negative nodes completely from the tree

B. Use max(0, child sums) to ignore negative paths and update global max with left + right + node.Val

C. Only consider paths from root to leaf nodes

D. Sum all node values regardless of sign and return the total

Solution

1. Step 1: Handle negative child sums carefully

   Use max(0, child sums) to ignore negative paths that reduce the total sum.

2. Step 2: Update global maximum path sum correctly

   Update the global max with the sum of left + right + current node value to consider paths passing through the node.

3. Final Answer:

   Use max(0, child sums) to ignore negative paths and update global max with left + right + node.Val -> Option B

4. Quick Check:

   Ignore negatives, update max with left+right+node [OK]

Hint: Ignore negative child sums, update max with combined path [OK]

Common Mistakes:

• Summing all nodes including negatives blindly
• Considering only root-to-leaf paths
• Removing negative nodes instead of ignoring sums