On the maximal size of tree in a random forest

Galton-Watson forests consisting of $N$ rooted trees and $n$ non-root vertices are considered. The distribution of the forest is determined by that of critical branching process with infinite variance and regularly varying tail of the progeny distribution. We prove limit theorem for the maximal size of a tree in a forest as $N,n \rightarrow \infty$ in such a way that $n/N \rightarrow \infty$. Our conditions are significantly wider than was previously known.