Cayley's formula

Take $n$ labelled points, named $1,2,\dots ,n$. A tree on them is a way of joining them with $n-1$ lines so that every point can be reached from every other. Two trees are different if they join different pairs.

Show that the number of trees is
$$n^{n-2}.$$

For $n=4$ this says there are $16$. Draw them all before you look for a proof.