- there are two ways to see if the target sum is in a sequence
- the first way is checking that the running sum on a node path equals the target sum
- the other way is if a subset of the path includes the target sum
- for the first way, we just keep track of the number of times the running total equaled the target sum
- for the second way, we keep track of the history of running totals for each path
- if we subtract the running total from the target sum, we can see what a previous running sum needed to be
- we check how many times this previous running sum actually happened and increment the count based on the number of times that running sum occured
- we make sure to decrement the running sum from the tracker after the recursive calls because we’re only using one tracker and we want it to track the running sums for only one path each time

# [4.12] Paths with Sum: You are given a binary tree in # which each node contains an integer value (which might # be positive or negative). Design an algorithm to count # 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) import unittest def count_paths(root, target_sum): running_total_tracker = {} running_total = 0 return count_paths_helper( root, target_sum, running_total_tracker, running_total ) def count_paths_helper(node, target_sum, tracker, running_total): if not node: return 0 matches = 0 running_total += node.value if running_total == target_sum: matches += 1 diff = running_total - target_sum if diff in tracker: matches += tracker if running_total in tracker: tracker[running_total] += 1 else: tracker[running_total] = 1 matches += count_paths_helper(node.left, target_sum, tracker, running_total) matches += count_paths_helper(node.right, target_sum, tracker, running_total) tracker[running_total] -= 1 return matches class TreeNode(object): def __init__(self, value, left=None, right=None): self.value = value self.left = left self.right = right class Test(unittest.TestCase): def setUp(self): self.root = TreeNode(1) self.n1 = TreeNode(2) self.n2 = TreeNode(3) self.n3 = TreeNode(1) self.n4 = TreeNode(1) self.n5 = TreeNode(-1) self.root.left = self.n1 self.root.right = self.n2 self.n1.right = self.n3 self.n2.right = self.n4 self.n4.right = self.n5 def test_count_paths(self): self.assertEqual(count_paths(self.root, 3), 4) self.n6 = TreeNode(3) self.n5.right = self.n6 self.assertEqual(count_paths(self.root, 3), 6) if __name__ == '__main__': unittest.main()