Use mathematical induction to prove that an undirected tree T of order n has n-1 edges
Proof:
When n=1, because the tree T has no loops, the number of edges of T is m=0 and m=n-1 holds.
It is true when n=k. When n=k+1, then T has at least two leaves. Let v be a leaf, delete v and its adjacent edges. At this time, the tree has n-1 vertices and m-1 edges. There is an induction hypothesis that (m − 1) = (n − 1) − 1. (m-1)=(n-1)-1.(m−1)=(n−1)−1 . I.e.,m = n - 1 m = nm=n−1 .
Proved.