Graph Theory: a tree and it's complement proof

[Magra2118]

Homework Statement

The complement of T' of a tree T with n vertices has [(n-1)(n-2)]/2 edges, for all integers n greater than or equal to 2.

Homework Equations

Must prove using induction and using Lemma 18.2 which states "any tree that has more than one vertex has at least one vertex of degree 1."

The Attempt at a Solution

Verify the basis case: p(2) [(2-1)(2-2)]/2 = 0, which is true because a tree with two vertices would have to have one edge, there its complement would have 0 edges

assume p(k) to be true: [(k-1)(k-2)]/2
show p(k+1) to be true: [(k+1-1)(k+1-2)]/2 = [(k)(k-1)]/2

Let G be a particular but arbitrarily chosen tree that has k vertices and k-1 edges. Let G' be the complement of tree G. By definition complement, G' have k vertices.

Don't know where to start/go after this!

 

[willem2]

A tree with k+1 edges would have a vertex of degree 1, because of the lemma. If you remove that you get a tree of k edges for which you assumed p(k) was true.

How many new edges will adding this vertex add to the complement?