Geodesic
On the number of trees of a~given size in a~Galton--Watson forest in the critical case
Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 1, pp. 75-92

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We consider a critical Galton–Watson branching process starting with $N$ particles and such that the number of offsprings of each particle is distributed as $p_k=(k+1)^{-\tau}-(k+2)^{-\tau}$, $k=0,1,2,\dots$ . For the corresponding Galton–Watson forest with $N$ trees and $n$ nonroot vertices, we find the limit distributions for the number of trees of a given size as $N,n \to \infty$, $n/ N^{\tau}\geq C>0$.
Keywords: Galton–Watson forest, number of trees of a given size
Mots-clés : limit distribution. 
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     title = {On the number of trees of a~given size in {a~Galton--Watson} forest in the critical case},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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E. V. Khvorostyanskaya. On the number of trees of a~given size in a~Galton--Watson forest in the critical case. Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 1, pp. 75-92. http://geodesic.mathdoc.fr/item/TVP_2023_68_1_a4/