number of labellings on a tree with given degrees

Statement

Let $d_{1}, \dots, d_{n}$ with each $d_{i} \geq 1$ and $\sum d_{i} = 2 (n - 1)$ . The number of labelled trees on $n$ vertices with the degree of the $i$ th vertex $d_{i}$ is equal to

(\binom{n - 2}{d_{1} - 1, d_{2} - 1, \dots, d_{n} - 1}) = \frac{(n - 2)!}{(d_{1} - 1)! (d_{2} - 1)! \dots (d_{n} - 1)!}

Proof

This is exactly the coefficient of $x_{1}^{d_{1}} \dots x_{n}^{d_{n}}$ in $F_{n}$ defined in proof 1 of Cayley's formula.

For proof 2, note that these correspond to codes $(c_{1}, \dots, c_{n - 2})$ such that each $i$ appears $d_{i} - 1$ times in the code.