Labelled tree count with prescribed degrees ($\frac{(n-2)!}{{\prod }_{v=1}^n(d_v-1)!}$)

For $n\ge 2$ and positive integers $d_1,\dots ,d_n$ with ${\sum }_v{d}_v=2(n-1)$, the number of labelled trees having vertex degree $d_v$ at label $v$ isBy the Prüfer degree formula, this is the multinomial coefficient counting sequences with $d_v-1$ occurrences of each label. If a degree is zero or the sum is wrong, there are no such trees. For the example degree sequence $(2,1,3,1,2,1)$, there are $4!/(1!0!2!0!1!0!)=12$ labelled trees, one for each ordering of the labels $1,3,3,5$. Knowing the vertex degrees thus constrains, but usually does not determine, the tree.