Asymptotic formulas for the enumerator of trees with a given number of hanging or internal vertices

V. A. Voblyi

Matematičeskie zametki, Tome 21 (1977) no. 1, pp. 65-70

Résumé

Let $t(r,n)$ be the number of trees with $n$ vertices of which $r$ are hanging and $q$ are internal ($r=n-q$). For a fixed $r$ or $q$ we prove the validity of the asymptotic formulas ($r>2$) \begin{gather*} t(r,n)\approx\frac1{r!(r-2)!}2^{2-r}n^{2r-4}\quad(n\to\infty), \\ t(n-q,n)\approx\frac1{q!(q-1)!}q^{q-2}n^{q-1}\quad(n\to\infty). \end{gather*} In the derivation of these formulas we do not use the expression for the enumerator of the trees with respect to the number of hanging vertices.