Path Sum III is a classic binary tree problem that appears frequently in coding interviews, especially when evaluating a candidate’s understanding of tree traversal and recursive problem-solving. The goal is to determine how many paths within a binary tree add up to a given target sum.
Each path must follow the parent-to-child direction, but unlike many tree path problems, the path does not need to start at the root or end at a leaf. A valid path can begin and end at any node, as long as it moves downward. This added flexibility is what makes Path Sum III more challenging than it initially appears.
You are given the root of a binary tree and a target sum.
Your task is to count the total number of distinct downward paths whose node values add up exactly to the target sum.
At first glance, this problem may seem like a straightforward tree traversal. However, many candidates quickly realize that simply checking all possible paths leads to repeated calculations and inefficient performance, especially for large or unbalanced trees.
From an interview perspective, Path Sum III tests tree traversal fundamentals, recursive thinking, the ability to optimize brute force solutions, and an understanding of prefix sums and cumulative state.
Why naive solutions fall short
A common initial approach is to start a depth-first search from every node and calculate all possible downward path sums. While this method works functionally, it is inefficient.
In the worst case, this approach can degrade to O(n²) time complexity, particularly for skewed trees where each node has only one child. Interviewers often let candidates describe this solution first, but they expect a more optimized strategy to follow.
The key realization is that many subpaths share partial sums. Recalculating them repeatedly wastes time and effort.
Want to explore more coding problem solutions? Check out the Repeated Substring Pattern and Minimum Add to Make Parentheses Valid coding problem solutions.
The optimized solution approach explained
The most efficient way to solve Path Sum III uses the concept of prefix sums, adapted from array-based problems to work in a tree structure.
Step 1: Track cumulative sums during traversal
As you traverse the tree from top to bottom, maintain a running total of the values along the current path. This running total represents the sum from the root of the traversal down to the current node.
At each node, you check whether there exists a previously seen cumulative sum such that:
current sum - previous sum = target sum
If such a sum exists, it means the path between those two points adds up to the target.
Step 2: Use a frequency map for efficiency
To make this check efficient, maintain a map that records how many times each cumulative sum has occurred along the current path. This allows you to count valid paths in constant time at each node.
When you move down the tree, you add the current cumulative sum to the map, recurse into the left and right children, and then remove the current sum when backtracking.
This careful bookkeeping ensures that only valid downward paths are counted and that sibling branches do not interfere with each other.
Why this approach is powerful
This prefix-sum-based solution runs in O(n) time, where n is the number of nodes in the tree. Each node is visited once, and all operations during traversal are constant time.
More importantly, it demonstrates a deeper understanding of problem-solving. It shows that you can recognize repeated substructures, apply ideas across different problem domains, and manage state correctly during recursion.
These are exactly the qualities interviewers look for when asking Path Sum III.
Common pitfalls to avoid
Candidates often make mistakes such as forgetting to backtrack the frequency map, confusing root-to-leaf paths with general downward paths, and double-counting overlapping paths.
Being able to explain how the prefix sum map is updated and cleaned up during recursion is just as important as arriving at the correct result.
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