Tree paths

A path is a route that connects two nodes, and the length of a path is the number of steps on the route. Your task is to compute the sum of the lengths of all paths connecting two leaves of a tree.

In the code template example, there are three such paths:

Here the path lengths are 3, 3, and 2, and so the correct answer is 3+3+2=8.

You may assume that the tree has at most 100 nodes.

In a file treepath.py, implement a function count that returns the sum of the lengths.

from collections import namedtuple

def count(node):
    # TODO

if __name__ == "__main__":
    Node = namedtuple("Node",["left","right"])
    tree = Node(Node(None,None),Node(Node(None,None),Node(None,None)))
    print(count(tree)) # 8