What is wrong with the following "proof" using mathematical induction of the statement that every tree with n vertices has a path of length n − 1. Basis step: Every tree with one vertex clearly has a path of length 0. Inductive step: Assume that a tree with n vertices has a path of length n − 1, which has u as its terminal vertex. Add a vertex v and the edge from u to v. The resulting tree has n + 1 vertices and has a path of length n. This completes the inductive step.