DP on Trees Maximum Path Sum in DSA Go - 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 Go code that calculates the maximum path sum in a binary tree?

DSA Go

package main
import "fmt"
type TreeNode struct {
    Val int
    Left *TreeNode
    Right *TreeNode
}

var maxSum int

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

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

func maxPathSum(root *TreeNode) int {
    maxSum = -1 << 31
    maxPathSumUtil(root)
    return maxSum
}

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))
}

A 35

B 20

C 42

D 27

💡 Hint

Consider the path that includes both left and right children of the node with value 20.

The maximum path sum is 42, coming from the path 15 -> 20 -> 7, plus the root's right subtree. The path includes nodes 15, 20, and 7, summing to 42.

❓ Predict Output

intermediate

Maximum Path Sum with Negative Values

What is the output of this Go program computing maximum path sum in a tree with negative values?

DSA Go

package main
import "fmt"
type TreeNode struct {
    Val int
    Left *TreeNode
    Right *TreeNode
}

var maxSum int

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

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

func maxPathSum(root *TreeNode) int {
    maxSum = -1 << 31
    maxPathSumUtil(root)
    return maxSum
}

func main() {
    root := &TreeNode{Val: -3}
    fmt.Println(maxPathSum(root))
}

A 3

B 0

C -1

D -3

💡 Hint

The tree has only one node with a negative value.

Since the tree has only one node with value -3, the maximum path sum is -3 itself.

🔧 Debug

advanced

Identify the Bug in Maximum Path Sum Calculation

What error will this Go code produce when calculating maximum path sum in a binary tree?

DSA Go

package main
import "fmt"
type TreeNode struct {
    Val int
    Left *TreeNode
    Right *TreeNode
}

var maxSum int

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

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

func maxPathSum(root *TreeNode) int {
    maxSum = 0
    maxPathSumUtil(root)
    return maxSum
}

func main() {
    root := &TreeNode{Val: -1}
    root.Left = &TreeNode{Val: -2}
    root.Right = &TreeNode{Val: -3}
    fmt.Println(maxPathSum(root))
}

A Returns 0 incorrectly

B Compilation error due to uninitialized maxSum

C Returns -1 correctly

D Runtime panic due to nil pointer dereference

💡 Hint

Check the initial value of maxSum before recursion.

maxSum is initialized to 0, so if all path sums are less than 0, maxSum remains 0. Here, the maximum path sum is -1, but since all current sums are negative, maxSum remains 0 and returns 0 incorrectly.

🧠 Conceptual

advanced

Understanding DP State in Maximum Path Sum on Trees

In the dynamic programming approach to find the maximum path sum in a binary tree, what does the returned value from the recursive helper function represent?

A The maximum path sum including both left and right children through the current node

B The maximum path sum of any path in the entire tree

C The maximum sum of a path starting from the current node and extending down to one child only

D The sum of values of all nodes in the subtree rooted at the current node

💡 Hint

Think about what the recursion returns to its parent call.

The recursive function 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 to compute larger paths.

🚀 Application

expert

Maximum Path Sum in a Tree with Multiple Negative and Positive Values

Given the following tree structure, what is the maximum path sum? Tree: - Root: 2 - Left child: -1 - Right child: 3 - Right child's left child: -2 - Right child's right child: 4 Use the maximum path sum definition where the path can start and end at any node.

A 7

B 9

C 6

D 5

💡 Hint

Consider the path 3 -> 4 and the root node 2.

The maximum path sum is 9, coming from the path 2 -> 3 -> 4. The negative values are excluded by taking max with 0 in the recursion.

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