Use the Matrix-Tree Formula to compute $\displaystyle \tau(K_{2,n}) $ for any $\displaystyle 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 $\displaystyle D(G)-A(G)= \tau(K_{2,1})=1 $ and the cofactors of $\displaystyle 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!