Let T be a tree. For any vertex v of T, we let cv denote the size of the largest component of the graph T∖v. (Recall that T∖v is the graph obtained from T by removing the vertex v and all edges that contain v. Note that a component (according to our definition) is a set of vertices; thus, its size is the number of vertices in it.)
The vertices v of T that minimize the number cv are called the centroids of T.
(a) Prove that T has no more than two centroids, and furthermore, if T has two centroids, then these two centroids are adjacent.
(b) Find a tree T such that the centroid(s) of T are distinct from the center(s) of T.