There are $n^{n-3}$ numbers of trees with named edges - how to proof?

Question

How to proof that there are

$$ n^{n-3} $$

trees with $n$ (unnamed) vertexes and $n-1$ named edges: $\left\{1, 2, 3, 4, ..., n-1\right\}$?

Answer 1

This is a consequence of a result from Cayley, known as…Cayley's formula.

Answer 2

HINT: There are $n$ ways to pick one vertex to be the root of the tree. Once you’ve done that, you can define a direction on each edge by considering its relationship to the root. Then use that directions of the edges together with their label to label the vertices other than the root. Then apply Cayley’s formula.