Let \(P\) be a path in \(T\) of longest possible length. Let \(u\) and \(v\) be the endpoints of the path. Since \(T\) does not have two vertices of degree one, at least one of these must have degree two or higher. Say that it is \(u\text{.}\) We know that \(u\) is adjacent to a vertex in the path \(P\text{,}\) but now it must also be adjacent to another vertex, call it \(u'\text{.}\)
in-context