On the number of trees of a given size in a Galton–Watson forest in the critical case

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 $N,n \to \infty$, $n/ N^{\tau}\geq C>0$.