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

1. LetTbe a tree andmthe length of the longest path inT.

Prove that the center ofTconsists of one point if and only ifmis even.

2. SupposeG=(V,E)a coherent weighed graph with weightfunctiong,for what counts that every weight is different.

Prove thatGconsists of exactly one spanning tree of minimum weight.