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Asymptotic formulas for the enumerator of trees with a given number of hanging or internal vertices
V. A. Voblyi State Scientific-Research and Design Institute for the Varnish and Paint Industry
Abstract:
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.
Received: 16.10.1975
Citation:
V. A. Voblyi, “Asymptotic formulas for the enumerator of trees with a given number of hanging or internal vertices”, Mat. Zametki, 21:1 (1977), 65–70; Math. Notes, 21:1 (1977), 36–39
Linking options:
https://www.mathnet.ru/eng/mzm7930 https://www.mathnet.ru/eng/mzm/v21/i1/p65
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Abstract page: | 201 | Full-text PDF : | 92 | First page: | 1 |
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