> ## Documentation Index
> Fetch the complete documentation index at: https://leetcode-py.wisl.dev/llms.txt
> Use this file to discover all available pages before exploring further.

> ## Agent Instructions
> leetcode-py is a Python LeetCode practice environment generator with one CLI: lcpy. It is not a service or platform.
> Each problem is a directory under leetcode/ with README.md, solution.py, test_solution.py, helpers.py, and playground.ipynb. lcpy gen creates them from JSON templates bundled with the package.
> Examples are backed by tests; copy them verbatim.

# Path Sum III Python Solution with Tests

> Tested Python solution for LeetCode 437 with 15 pytest cases. Generate a practice environment with lcpy.

LeetCode 437, Medium. Topics: Tree, Depth-First Search, Binary Tree. [View on LeetCode](https://leetcode.com/problems/path-sum-iii/description/).

Generate this problem as a practice environment: tested reference solution, 15 [parametrized pytest cases](/practice/testing), and a playground notebook:

```bash theme={"theme":{"light":"github-light","dark":"github-dark"}}
lcpy gen -n 437   # by problem number
lcpy gen -s path_sum_iii   # by problem name
```

## Problem

Given the `root` of a binary tree and an integer `targetSum`, return *the number of paths where the sum of the values along the path equals* `targetSum`.

The path does not need to start or end at the root or a leaf, but it must go downwards (i.e., traveling only from parent nodes to child nodes).

### Examples

![Example 1](https://assets.leetcode.com/uploads/2021/04/09/pathsum3-1-tree.jpg)

```
Input: root = [10,5,-3,3,2,None,11,3,-2,None,1], targetSum = 8
Output: 3
Explanation: The paths that sum to 8 are shown.
```

```
Input: root = [5,4,8,11,None,13,4,7,2,None,None,5,1], targetSum = 22
Output: 3
```

### Constraints

* The number of nodes in the tree is in the range \[0, 1000].
* -10^9 \<= Node.val \<= 10^9
* -1000 \<= targetSum \<= 1000

## Solution

Reference implementation from [solution.py on GitHub](https://github.com/wislertt/leetcode-py/blob/main/leetcode/path_sum_iii/solution.py), full suite in [test\_solution.py](https://github.com/wislertt/leetcode-py/blob/main/leetcode/path_sum_iii/test_solution.py):

```python theme={"theme":{"light":"github-light","dark":"github-dark"}}
from collections import defaultdict

from leetcode_py import TreeNode


class Solution:
    # Time: O(n)
    # Space: O(n)
    def path_sum(self, root: TreeNode[int] | None, target_sum: int) -> int:
        """Count paths where sum equals target_sum using prefix sum technique."""
        self.count = 0
        prefix_counts: defaultdict[int, int] = defaultdict(int)
        prefix_counts[0] = 1  # Empty path prefix

        def dfs(node: TreeNode[int] | None, current_sum: int) -> None:
            if not node:
                return

            current_sum += node.val
            # Check if (current_sum - target_sum) exists in prefix_counts
            self.count += prefix_counts[current_sum - target_sum]
            # Add current sum to prefix_counts
            prefix_counts[current_sum] += 1

            # Recurse on children
            dfs(node.left, current_sum)
            dfs(node.right, current_sum)

            # Backtrack: remove current sum from prefix_counts
            prefix_counts[current_sum] -= 1

        dfs(root, 0)
        return self.count
```

## Complexity

| Time | Space |
| ---- | ----- |
| O(n) | O(n)  |

## Tags

[Grind](/catalog/grind), [AlgoMaster 75](/catalog/algo-master-75).