Geodesic
Asymptotic formulas for the enumerator of trees with a~given number of hanging or internal vertices
Matematičeskie zametki, Tome 21 (1977) no. 1, pp. 65-70.

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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. 
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     author = {V. A. Voblyi},
     title = {Asymptotic formulas for the enumerator of trees with a~given number of hanging or internal vertices},
     journal = {Matemati\v{c}eskie zametki},
     pages = {65--70},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_21_1_a7/}
}
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V. A. Voblyi. Asymptotic formulas for the enumerator of trees with a~given number of hanging or internal vertices. Matematičeskie zametki, Tome 21 (1977) no. 1, pp. 65-70. http://geodesic.mathdoc.fr/item/MZM_1977_21_1_a7/