Introduction

Maximum Path Sum means:

  • finding path
    with maximum total value
    inside binary tree

The path:

  • can start anywhere
  • can end anywhere
  • must remain connected

Goal:

  • maximize total path sum

Example:

        -10
       /    \
      9      20
            /  \
           15   7


Maximum Path:
15 → 20 → 7

Maximum Sum:
42

Explanation:

The best path does not always pass through root node.

This problem is one of the most important applications of:

DFS Traversal

Constraints

 1 <= Number of Nodes <= 10^5

Approach : Recursive DFS Solution

Explanations:

Explanation:

The idea is:

  • recursively calculate
    maximum contribution
    from every subtree
  • update global maximum sum

Steps:

  1. Visit current node.
  2. Calculate left contribution.
  3. Calculate right contribution.
  4. Ignore negative paths.
  5. Update maximum sum.
  6. Return best subtree path.

Formula:

node.value + leftContribution + rightContribution

Return Formula:

node.value + max(leftContribution, rightContribution)

This approach:

  • uses DFS recursion
  • combines subtree optimization
    and path calculation

Dry Run

Visit:20

Left contribution:
15

Right contribution:
7

Current path sum:
42

Update maximum sum:
42

Return:
35

Practice :

Complexity Analysis :

Time Complexity:- O(n)Explanation :
Every tree node is visited once.

Space Complexity:- O(h)
Explanation :
Recursion stack depends on tree height.

Why This Problem is Important

This problem builds the foundation for:

  • DFS traversal
  • Tree Dynamic Programming
  • Recursive optimization
  • Path calculations
  • Binary tree analysis

Real-World Applications

Maximum path concepts are used in:

  • Network optimization
  • Route analysis
  • Financial systems
  • AI decision trees
  • Graph optimization systems

Common Beginner Mistakes

  • Including negative paths
  • Incorrect global update
  • Wrong return value
  • Missing recursion logic
  • Confusing path and subtree contribution

Interview Tip

Interviewers often expect:

  • DFS understanding
  • recursion explanation
  • subtree contribution logic
  • optimization discussion

Always explain:

  • negative path removal
  • contribution calculation
  • global maximum update

Related Questions

  • Path Sum II
  • Root to Leaf Paths
  • Diameter of Binary Tree
  • DFS Traversal
  • Binary Tree Paths

Final Takeaway

The Maximum Path Sum problem is one of the most important intermediate DFS tree problems.

It teaches:

  • DFS recursion
  • tree dynamic programming
  • path optimization
  • recursive contribution calculation

Understanding this problem builds a strong foundation for:

  • advanced tree problems
  • graph algorithms
  • interview-level optimization problems.