Ok, I have 2 questions.

1) So the center of a tree is defined as the set of vertices that have the smallest eccentricity (meaning the number of edges in a longest path that begins at a vertex x). How do you prove by induction that every tree has a center that consists of either one or two vertices?

2) Also I was wondering if T is a tree that has more than one edge, how do you prove that the center of T equals the center of p(T)? p(T) is defined as the tree that has all its leaves (leaves are vertices of degree 1) taken out (as well as the edges attached to those degree 1 vertices).

Thanx to anyone who can help!