Ghoul-icious Goodies

*Despite my ghoulish reputation, I really have the heart of a small boy. I keep it in a jar on my desk.* Alice wants to give Bob some goodies, but she's uncertain about how many to give. To help determine the quantity, she presented Bob with the following challenge: Alice has a tree, i.e, a simple connected undirected graph with $N$ vertices and $N-1$ edges. Each edge has a lowercase character w