Number of spanning trees for $K_n-e$

Question

As the title suggests, I want to calculate the number of spanning trees in $K_n - e$ where $K_n$ is the complete graph on $n$ vertices and $e$ is any edge. The answer to this problem is $(n-2)*(n)^{n-3}$ and i know one method to do this. My actual question is regarding one of my approaches which is wrong. So let's pick any vertex of the graph, say $v_1$ and delete any edge for it. now it has $n-2$ edges and through this i can reach any of the vertex. Now all the remaining vertices form a $K_{n-1}$ graph and these would have $(n-1)^{n-3}$ spanning trees so the total spanning trees would be $(n-2)*(n-1)^{n-3}$. Where am i exactly going wrong and is it possible to proceed with this approach? Thank you

Answer 1

Your method counts only the spanning trees where $v_1$ is a leaf. A spanning tree of $K_n - e$ does not necessarily consist of a spanning tree of $K_n - v_1$ plus an edge to $v_1$ other than $e$.