Given n numbered vertices I want to know the number of different trees that can be created with them. I know that cayley's theorem says it's $n^{n-2}$, but why can't it also be: $\binom{\binom{n}{2}}{n-1}$
Since $\binom{n}{2}$ is the number of edges out of which I choose $n-1$ (which makes it a graph with $n$ nodes and $n-1$ edges which is a tree.