Abstract:
Galton-Watson forests formed by a critical branching process starting with $N$ particles are considered. The total number of descendants of the initial particles is equal to $n$ untill extinction. The number of offspring of each particle has the distribution
\begin{equation*}p_k=\frac{h(k+1)}{(k+1)^\tau}, \quad k=0,1,2, \dots, \quad \tau\in (2,3),\end{equation*}
where $h(x)$ is a slowly varying at infinity function such that $h(x)\rightarrow D, 0<D<\infty,$ as $x\rightarrow \infty$. The limit theorems on the number of trees of a given size and on the maximum size of a tree are proved as $N,n\rightarrow \infty$ in such a way that $n/N \rightarrow \infty$ and $n/N^{\tau-1} \rightarrow 0$.
Keywords:Galton-Watson forest, tree size, limit theorems.
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