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

Recall & Review

beginner

What is the main goal of the "Maximum Path Sum" problem on trees?

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

intermediate

In DP on trees for maximum path sum, what does the "return value" of the recursive function usually represent?

The maximum sum of a path starting from the current node and extending downwards to one of its children or just the node itself.

intermediate

Why do we consider both left and right child paths when calculating maximum path sum at a node?

Because the maximum path might pass through the current node connecting left and right subtrees, forming a path that goes down left child, up to node, then down right child.

beginner

What is the role of a global variable in the DP on trees maximum path sum solution?

It keeps track of the overall maximum path sum found so far during the recursion, since the maximum path might be anywhere in the tree.

intermediate

How do negative node values affect the maximum path sum calculation?

Negative values can reduce the sum, so we ignore paths with negative sums by taking max with zero to avoid decreasing the total.

In the maximum path sum problem on trees, what does the path represent?

A Any set of nodes in the tree

B A sequence of connected nodes where each node is visited once

C Only paths from root to leaf

D Only paths from leaf to leaf

The path is a sequence of connected nodes without revisiting nodes, and can start and end anywhere.

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

A Sum of left child path + node + right child path

B Maximum path sum from right child

C Sum of node and its grandparent

D Maximum path sum from left child

The path must be continuous, so grandparent skipping is not allowed.

Why do we take max with zero when calculating child path sums in maximum path sum problem?

A To count zero nodes

B To include negative values always

C To avoid overflow

D To ignore negative sums that reduce total

Negative sums reduce the total, so we ignore them by taking max with zero.

What data structure is typically used to represent the tree in maximum path sum problems?

A Binary tree nodes with left and right pointers

B Linked list

C Hash map

D Array

Binary trees are represented with nodes having left and right child pointers.

What is the time complexity of the DP on trees maximum path sum solution?

A O(n)

B O(n^2)

C O(log n)

D O(n log n)

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

Explain how dynamic programming is applied on trees to find the maximum path sum.

Think about how you combine results from children and update a global answer.

Describe the role of the global variable in the maximum path sum problem on trees and why it is necessary.

Why can't the recursive function return the final answer alone?

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