The emergence of a giant tree in a random forest

We consider random forests consisting of $N$ trees of a simply generated family of rooted trees with $n$ non-root vertices. For these forests, we investigate the conditions when the giant tree comes into existence in the random forest and demonstrate that the giant component can occur only in the case where $N,n\to\infty $ so that $n/N^2\to\infty$.
This research was supported by the Russian Foundation for Basic Research, grant 97–01–00065.