Maximum Path Sum in Binary Tree in DSA C - Time & Space Complexity

Time Complexity: Maximum Path Sum in Binary Tree

O(n)

Understanding Time Complexity

We want to understand how the time needed to find the maximum path sum in a binary tree grows as the tree gets bigger.

Specifically, how does the number of steps change when the number of nodes increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

int maxPathSumHelper(struct TreeNode* root, int* maxSum) {
    if (root == NULL) return 0;
    int left = maxPathSumHelper(root->left, maxSum);
    int right = maxPathSumHelper(root->right, maxSum);
    int maxSingle = fmax(fmax(left, right) + root->val, root->val);
    int maxTop = fmax(maxSingle, left + right + root->val);
    *maxSum = fmax(*maxSum, maxTop);
    return maxSingle;
}

int maxPathSum(struct TreeNode* root) {
    int maxSum = INT_MIN;
    maxPathSumHelper(root, &maxSum);
    return maxSum;
}
    

This code finds the maximum path sum in a binary tree by visiting each node once and calculating the best path through it.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls visiting each node once.
• How many times: Each node is visited exactly one time.

How Execution Grows With Input

As the number of nodes (n) grows, the function visits each node once, doing a fixed amount of work per node.

Input Size (n)	Approx. Operations
10	About 10 visits
100	About 100 visits
1000	About 1000 visits

Pattern observation: The number of operations grows directly with the number of nodes.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the maximum path sum grows linearly with the number of nodes in the tree.

Common Mistake

[X] Wrong: "The recursion visits some nodes multiple times, so time is more than linear."

[OK] Correct: Each node is visited once in the recursion, and results are combined on the way back, so no repeated work happens.

Interview Connect

Understanding this linear time complexity helps you explain how efficient your solution is and shows you can analyze recursive tree problems clearly.

Self-Check

"What if the tree was very unbalanced and looked like a linked list? How would the time complexity change?"

Practice

1. What does the "maximum path sum" in a binary tree represent?

easy

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

B. The sum of all node values in the tree

C. The largest sum of values along any path between two nodes

D. The sum of values from root to the deepest leaf only

Solution

1. Step 1: Understand the path definition

   The path can start and end at any two nodes, not necessarily root or leaf.

2. Step 2: Identify maximum sum meaning

   It is the largest sum of node values along any such path in the tree.

3. Final Answer:

   The largest sum of values along any path between two nodes -> Option C

4. Quick Check:

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

Hint: Maximum path sum is any path, not just root to leaf [OK]

Common Mistakes:

• Thinking it must start at root
• Confusing with sum of all nodes
• Considering only leaf nodes

2. Which of the following C function signatures is correct for a recursive function to find maximum path sum in a binary tree node defined as struct TreeNode?

easy

A. int maxPathSum(struct TreeNode* root);

B. void maxPathSum(struct TreeNode root);

C. int maxPathSum(struct TreeNode root);

D. int maxPathSum(struct TreeNode** root);

Solution

1. Step 1: Understand parameter passing for trees in C

   Binary trees are usually passed as pointers to nodes, so struct TreeNode* is correct.

2. Step 2: Check return type and parameter correctness

   Function returns int (maximum sum), and parameter is pointer to root node, so int maxPathSum(struct TreeNode* root); is correct.

3. Final Answer:

   int maxPathSum(struct TreeNode* root); -> Option A

4. Quick Check:

   Use pointer parameter and int return type [OK]

Hint: Use pointer to struct for tree nodes in C functions [OK]

Common Mistakes:

• Passing struct by value instead of pointer
• Using void return type instead of int
• Using double pointer unnecessarily

3. Given the binary tree below, what is the maximum path sum?

    1
   / \
  2   3

medium

A. 5

B. 6

C. 4

D. 3

Solution

1. Step 1: Identify all possible paths and their sums

   Paths: 2->1->3 sum = 6, 1->2 sum = 3, 1->3 sum = 4, single nodes 2,3,1.

2. Step 2: Find the maximum sum among these paths

   Maximum sum is 6 from path 2->1->3.

3. Final Answer:

   6 -> Option B

4. Quick Check:

   Sum of 2 + 1 + 3 = 6 [OK]

Hint: Check paths through root connecting left and right [OK]

Common Mistakes:

• Considering only root to leaf paths
• Ignoring paths that go through root connecting children
• Choosing max node value instead of path sum

4. Consider this buggy C code snippet for maximum path sum calculation:

int maxPathSumUtil(struct TreeNode* root, int* maxSum) {
    if (!root) return 0;
    int left = maxPathSumUtil(root->left, maxSum);
    int right = maxPathSumUtil(root->right, maxSum);
    int maxSingle = max(max(left, right) + root->val, root->val);
    int maxTop = max(maxSingle, left + right + root->val);
    *maxSum = max(*maxSum, maxTop);
    return maxSingle;
}

What is the main bug in this code?

medium

A. It does not update maxSum pointer correctly

B. It does not handle negative node values correctly

C. It returns maxTop instead of maxSingle

D. It uses max function without defining it

Solution

1. Step 1: Check usage of max function

   The code uses max() but C standard library does not have max function by default; it must be defined or replaced.

2. Step 2: Verify other logic parts

   Return and pointer update logic is correct; negative values handled by max comparisons.

3. Final Answer:

   It uses max function without defining it -> Option D

4. Quick Check:

   max() must be defined or replaced in C [OK]

Hint: Remember C has no built-in max function [OK]

Common Mistakes:

• Assuming max() is built-in in C
• Confusing return values with maxSum update
• Ignoring pointer usage correctness

5. You want to find the maximum path sum in a binary tree where some nodes have negative values. Which approach correctly handles negative values to find the maximum path sum?

hard

A. Use recursion to compute max path sums including negative values and update global max accordingly

B. Calculate sum of all nodes and subtract minimum negative node value

C. Only consider paths that include positive nodes and skip negative nodes completely

D. Ignore negative values by treating them as zero during recursion

Solution

1. Step 1: Understand handling of negative values in max path sum

   Negative values can reduce path sums, so recursion must consider them but choose max sums carefully.

2. Step 2: Use recursion with max comparisons to update global max

   Recursion returns max single path sum including negative values, updating global max with max of left+right+node or single path sums.

3. Final Answer:

   Use recursion to compute max path sums including negative values and update global max accordingly -> Option A

4. Quick Check:

   Include negatives but choose max sums carefully [OK]

Hint: Include negatives but pick max sums in recursion [OK]

Common Mistakes:

• Ignoring negative nodes completely
• Treating negatives as zero forcibly
• Using simple sums without max comparisons