Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / R / R2 / v / b

First choose the $l$ absent labels, which will be the leaves of a graph, in $\left(\genfrac{}{}{0ex}{}{n}{l}\right)$ ways. Every other label must occur exactly $r-1$ times in the Prüfer sequence. There are $n-l$ such labels and their total multiplicity is $n-2$ by (a). The number of orderings of this multiset is $(n-2)!/((r-1)!{)}^{n-l}$. The bijection in (i) turns each ordering into one distinct labelled tree and exhausts all possibilities. ThereforeThe formula is used with the feasible integer parameters from (a), and gives one for the two-vertex edge with no internal labels.