Limit theorems for the maximal tree size of a Galton – Watson forest in the critical case

We consider a critical Galton – Watson branching process starting with $N$ particles; the number of offsprings is supposed to have the distribution $p_k=(k+1)^{-\tau}-(k+2)^{-\tau}$, $k=0,1,2,\ldots$ Limit distributions of the maximal tree size are obtained for the corresponding Galton – Watson forest with $N$ trees and $n$ non-root vertices as $N,n\to\infty$, $n/N^{\tau}\geq C> 0$.