Fixed point of tree automorphism Given the tree $T$ and its automorphism $\phi$ prove that there exists a vertex $v$ such that $\phi(v)=v$ or an edge $\{{u,v\}}$ such that $\phi(\{{u,v}\}) = \{{u,v}\}$

See this wikipedia article. The center is preserved by any automorphism. There is also the baricenter of a tree (which is a vertex for which the sum of the distances to the other points is minimized). Again, there is either one such or two adjacent ones. Are the center and the baricenter the same? For you to find out.
