# 30. Path Sum III

You are given a binary tree in which each node contains an integer value.

Find the number of paths that sum to a given value.

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

The tree has no more than 1,000 nodes and the values are in the range -1,000,000 to 1,000,000.

**Example:**

```
root = [10,5,-3,3,2,null,11,3,-2,null,1], sum = 8

      10
     /  \
    5   -3
   / \    \
  3   2   11
 / \   \
3  -2   1

Return 3. The paths that sum to 8 are:

1.  5 -> 3
2.  5 -> 2 -> 1
3. -3 -> 11
```

## Solution: (Using Prefix Sum on tree)

```cpp
class Solution
{
public:
    unordered_map<int, int> m;
    int count = 0;
    
    void traverse(TreeNode *root, int sum, int target)
    {

        if (root == NULL)
        {
            return;
        }

        int s = sum + root->val;
        
        if (s == target)
        {
            count++;
        }

        if (m.find(s - target) != m.end())
        {
            count += m[s - target];
        }

        m[s]++;

        traverse(root->left, s, target);
        traverse(root->right, s, target);
        m[s]--;

        return;
    }

    int pathSum(TreeNode *root, int sum)
    {

        if (root == NULL)
        {
            return 0;
        }

        traverse(root, 0, sum);

        return count;
    }
};
```

**Time Complexity: O(n)**
