Limit theorems for the number of trees of a given size in a Galton–Watson forest with a bounded number of vertices

We consider a random forest formed by trajectories of a homogeneous Galton – Watson branching process starting with $N$ particles, the number of offsprings has the distribution $p_k=(k+1)^{-\tau}-(k+2)^{-\tau}$, $k=0,1,2,\dots.$ For the Galton – Watson forest containing at most $n$ vertices limit distributions of the number of trees of a given size as $N,n\to\infty$ are obtained for different values of the parameter $\tau$.