Use the Matrix-Tree Formula to compute  \tau(K_{2,n}) for any  n \geq 1 . Can you find a combinatorial argument for your answer?

This is for a Graph Theory class, and I approached the problem by first finding that the cofactors of  D(G)-A(G)= \tau(K_{2,1})=1 and the cofactors of  D(G)-A(G)= \tau(K_{2,2})= 4 .

But once the matrices started to get larger, I had trouble finding determinants and wondered if I was even on the right track or if there is a much simpler way to find these cofactors. Any help would be much appreciated!