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

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Maximum Path Sum in Binary Tree

What is the output of the following C# code that calculates the maximum path sum in a binary tree?

DSA C#

public class TreeNode {
    public int val;
    public TreeNode left, right;
    public TreeNode(int val=0, TreeNode left=null, TreeNode right=null) {
        this.val = val;
        this.left = left;
        this.right = right;
    }
}

public class Solution {
    private int maxSum = int.MinValue;
    public int MaxPathSum(TreeNode root) {
        MaxGain(root);
        return maxSum;
    }
    private int MaxGain(TreeNode node) {
        if (node == null) return 0;
        int leftGain = Math.Max(MaxGain(node.left), 0);
        int rightGain = Math.Max(MaxGain(node.right), 0);
        int priceNewpath = node.val + leftGain + rightGain;
        maxSum = Math.Max(maxSum, priceNewpath);
        return node.val + Math.Max(leftGain, rightGain);
    }
}

// Tree structure:
//       -10
//       /  \
//      9   20
//          / \
//         15  7

var root = new TreeNode(-10,
    new TreeNode(9),
    new TreeNode(20, new TreeNode(15), new TreeNode(7))
);
var sol = new Solution();
System.Console.WriteLine(sol.MaxPathSum(root));

A 42

B 20

C 27

D 35

💡 Hint

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

The maximum path sum is the path 15 -> 20 -> 7, plus the node -10 is negative so not included in the max path. The path sum is 15 + 20 + 7 = 42.

🧠 Conceptual

intermediate

Understanding the Role of MaxGain Function

In the maximum path sum algorithm on a binary tree, what does the MaxGain function return for each node?

A The sum of the node's value plus both left and right children's maximum gains.

B The maximum sum of any path in the entire tree including the node.

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

D The minimum path sum from the node to any leaf.

💡 Hint

Think about what the function returns to its parent call.

MaxGain returns the maximum sum of a path starting at the current node and extending down to either left or right child (or none if negative). It helps to build paths upwards.

🔧 Debug

advanced

Identify the Bug in Maximum Path Sum Code

What error will occur when running this modified maximum path sum code?

DSA C#

public class Solution {
    private int maxSum = int.MinValue;
    public int MaxPathSum(TreeNode root) {
        MaxGain(root);
        return maxSum;
    }
    private int MaxGain(TreeNode node) {
        if (node == null) return 0;
        int leftGain = MaxGain(node.left);
        int rightGain = MaxGain(node.right);
        int priceNewpath = node.val + leftGain + rightGain;
        maxSum = Math.Max(maxSum, priceNewpath);
        return node.val + Math.Max(leftGain, rightGain);
    }
}

// Tree with negative values
var root = new TreeNode(-2, null, new TreeNode(-3));
var sol = new Solution();
System.Console.WriteLine(sol.MaxPathSum(root));

A Returns -3 as maximum path sum

B Returns -5 as maximum path sum

C Returns 0 as maximum path sum

D Returns -2 as maximum path sum

💡 Hint

Check how negative gains are handled in the recursion.

Without ignoring negative gains (no Math.Max with 0), the function sums negative values directly, so the path sum is -2 + 0 + -3 = -5.

📝 Syntax

advanced

Syntax Error in Recursive Maximum Path Sum Function

Which option contains a syntax error in the MaxGain function implementation?

A

private int MaxGain(TreeNode node) {
    if (node == null) return 0;
    int leftGain = Math.Max(MaxGain(node.left), 0);
    int rightGain = Math.Max(MaxGain(node.right), 0);
    int priceNewpath = node.val + leftGain + rightGain;
    maxSum = Math.Max(maxSum, priceNewpath);
    return node.val + Math.Max(leftGain, rightGain);
}

B

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C

rivate int MaxGain(TreeNode node) {
    if (node == null) return 0;
    int leftGain = Math.Max(MaxGain(node.left), 0);
    int rightGain = Math.Max(MaxGain(node.right), 0);
    int priceNewpath = node.val + leftGain + rightGain;
    maxSum = Math.Max(maxSum, priceNewpath);
    return node.val + Math.Max(leftGain, rightGain);
}

D

private int MaxGain(TreeNode node) {
    if (node == null) return 0
    int leftGain = Math.Max(MaxGain(node.left), 0);
    int rightGain = Math.Max(MaxGain(node.right), 0);
    int priceNewpath = node.val + leftGain + rightGain;
    maxSum = Math.Max(maxSum, priceNewpath);
    return node.val + Math.Max(leftGain, rightGain);
}

💡 Hint

Look carefully at the line endings.

Option D is missing a semicolon after the return 0 statement, causing a syntax error.

🚀 Application

expert

Maximum Path Sum in a Tree with Negative and Positive Values

Given the tree below, what is the maximum path sum computed by the DP on Trees algorithm?

DSA C#

// Tree structure:
//        2
//       / \
//     -1   3
//     /   / \
//   -4   5  -6
//        /
//       1

var root = new TreeNode(2,
    new TreeNode(-1, new TreeNode(-4), null),
    new TreeNode(3, new TreeNode(5, new TreeNode(1), null), new TreeNode(-6))
);
var sol = new Solution();
System.Console.WriteLine(sol.MaxPathSum(root));

A 11

B 10

C 9

D 8

💡 Hint

Consider the path 1 -> 5 -> 3 -> 2.

The maximum path sum is 1 + 5 + 3 + 2 = 11. Negative nodes are avoided in the path.

Practice

1. What does the "maximum path sum" in a tree represent in the context of DP on Trees Maximum Path Sum?

easy

A. The sum of all node values in the tree

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

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

D. The longest path length in terms of number of edges

Solution

1. Step 1: Understand the problem definition

   The maximum path sum means the highest total sum of node values collected along any path connecting two nodes in the tree, not necessarily passing through the root.

2. Step 2: Differentiate from other sums

   It is not the sum of all nodes or just the maximum single node value, but the sum along a path that can include multiple nodes.

3. Final Answer:

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

4. Quick Check:

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

Hint: Maximum path sum is about sums along paths, not single nodes [OK]

Common Mistakes:

• Confusing maximum path sum with sum of all nodes
• Thinking it is just the maximum node value
• Assuming path must start at root

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

easy

A. int MaxPathSum(TreeNode node)

B. void MaxPathSum(TreeNode node)

C. int MaxPathSum(int node)

D. TreeNode MaxPathSum(TreeNode node)

Solution

1. Step 1: Identify the return type needed

   The function should return an integer representing the maximum path sum starting from the given node.

2. Step 2: Check parameter type

   The parameter must be a TreeNode object to access node values and children.

3. Final Answer:

   int MaxPathSum(TreeNode node) -> Option A

4. Quick Check:

   Return type int and parameter TreeNode required [OK]

Hint: Return int for sums, input parameter must be TreeNode [OK]

Common Mistakes:

• Using void return type when int is needed
• Passing int instead of TreeNode as parameter
• Returning TreeNode instead of int

3. Consider the following C# code snippet for a binary tree node class and a method to find maximum path sum:

public class TreeNode {
    public int val;
    public TreeNode left, right;
    public TreeNode(int x) { val = x; }
}

int maxSum = int.MinValue;

int MaxPathSum(TreeNode node) {
    if (node == null) return 0;
    int left = Math.Max(0, MaxPathSum(node.left));
    int right = Math.Max(0, MaxPathSum(node.right));
    maxSum = Math.Max(maxSum, left + right + node.val);
    return Math.Max(left, right) + node.val;
}

// Tree structure:
//       1
//      / \
//     2   3

TreeNode root = new TreeNode(1);
root.left = new TreeNode(2);
root.right = new TreeNode(3);
MaxPathSum(root);
Console.WriteLine(maxSum);

What is the output printed?

medium

A. 1

B. 5

C. 3

D. 6

Solution

1. Step 1: Calculate max path sums for children

   For node 2 and 3, left and right children are null, so returns 0. Max path sum at node 2 and 3 is their own value (2 and 3).

2. Step 2: Calculate max path sum at root

   At root (1), left = max(0, 2) = 2, right = max(0, 3) = 3. maxSum updated to max(-∞, 2+3+1) = 6. Return max(2,3)+1=4.

3. Final Answer:

   6 -> Option D

4. Quick Check:

   Sum of path 2->1->3 = 6 [OK]

Hint: Add left + right + node.val for max path at each node [OK]

Common Mistakes:

• Ignoring max(0, child) and using negative sums
• Returning sum without updating maxSum
• Confusing return value with maxSum

4. The following C# code attempts to compute the maximum path sum in a binary tree but has a bug:

int maxSum = int.MinValue;

int MaxPathSum(TreeNode node) {
    if (node == null) return 0;
    int left = MaxPathSum(node.left);
    int right = MaxPathSum(node.right);
    maxSum = Math.Max(maxSum, left + right + node.val);
    return Math.Max(left, right) + node.val;
}

What is the bug in this code?

medium

A. It does not ignore negative sums from children

B. It returns the wrong value type

C. It does not update maxSum correctly

D. It does not handle null nodes

Solution

1. Step 1: Analyze child sums usage

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

2. Step 2: Identify fix for negative sums

   We must use Math.Max(0, left) and Math.Max(0, right) to ignore negative sums from children.

3. Final Answer:

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

4. Quick Check:

   Negative child sums must be ignored [OK]

Hint: Use Math.Max(0, childSum) to ignore negatives [OK]

Common Mistakes:

• Not using Math.Max(0, child) causing wrong sums
• Assuming return type is wrong
• Missing null checks (which are present)

5. Given a binary tree where some nodes have negative values, which approach correctly computes the maximum path sum using DP on trees? A) At each node, compute max path sum including both children and update global max; return max path sum including one child plus node value. B) At each node, sum all child node values and return total sum. C) At each node, return the maximum child value only. D) At each node, ignore children and consider only node value.

hard

A. Return the maximum child value only

B. Sum all child node values and return total sum

C. Compute max path sum including both children and update global max; return max path sum including one child plus node value

D. Ignore children and consider only node value

Solution

1. Step 1: Understand the correct DP approach

   The correct approach is to consider the maximum path sum including both children and update a global max, returning the max path sum including one child plus node value.

2. Step 2: Identify the incorrect options

   The other options do not correctly handle the problem. They either sum all children, return only the maximum child value, or ignore children and consider only node value.

3. Final Answer:

   Compute max path sum including both children and update global max; return max path sum including one child plus node value -> Option C

4. Quick Check:

   Correct DP uses both children sums and global max update [OK]

Hint: Include both children sums and update global max [OK]

Common Mistakes:

• Ignoring children nodes in calculation
• Summing all children without max checks
• Returning only child values without node