I've 2 questions and I'll be very grateful if someone could help me!!!!

1. Let T be a tree and m the length of the longest path in T.
Prove that the center of T consists of one point if and only if m is even.

2. Suppose G=(V,E) a coherent weighed graph with weightfunction g,for what counts that every weight is different.
Prove that G consists of exactly one spanning tree of minimum weight.