<p>Given the <code>root</code> of a binary tree and an integer <code>targetSum</code>, return <em>the number of paths where the sum of the values&nbsp;along the path equals</em>&nbsp;<code>targetSum</code>.</p>

<p>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).</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/09/pathsum3-1-tree.jpg" style="width: 450px; height: 386px;" />
<pre>
<strong>Input:</strong> root = [10,5,-3,3,2,null,11,3,-2,null,1], targetSum = 8
<strong>Output:</strong> 3
<strong>Explanation:</strong> The paths that sum to 8 are shown.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> root = [5,4,8,11,null,13,4,7,2,null,null,5,1], targetSum = 22
<strong>Output:</strong> 3
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li>The number of nodes in the tree is in the range <code>[0, 1000]</code>.</li>
	<li><code>-10<sup>9</sup> &lt;= Node.val &lt;= 10<sup>9</sup></code></li>
	<li><code>-1000 &lt;= targetSum &lt;= 1000</code></li>
</ul>

# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right
class Solution:
    def pathSum(self, root: Optional[TreeNode], targetSum: int) -> int:
        if not root:
            return 0

        # Traverse BFS
        # Create list of values seen
        # When we add new node: 
        #   - Add to DP table of sums of subarrays
        #   - Recalculate the last column of DP table
        # When we leave node:
        #   - Ignore DP table beyond depth

        s = [[root, 0]]
        sums = [[root.val]]
        sums_depth = 0
        ret = 0
        
        while s:
            node, depth = s.pop()
            # Make sure sums is big enough
            if depth > sums_depth:
                for i in range(depth):
                    sums[i].append(0)
                sums.append([0 for x in range(depth+1)])
                sums_depth = depth

            # Add our new value
            if node.val == targetSum:
                ret += 1
            sums[depth][depth] = node.val

            # Fill in missing sums
            for row in range(depth-1, -1, -1):
                sums[row][depth] = sums[row][depth-1] + sums[row+1][depth] - sums[row+1][depth-1]
                if sums[row][depth] == targetSum:
                    ret += 1

            # Add children
            if node.right:
                s.append([node.right, depth+1])
            if node.left:
                s.append([node.left, depth+1])

        return ret
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