The asymptotic distribution of maximum tree size in a random forest

A random forest containing $N+n$ points, of which $N$ points are roots, is considered. All the points are labelled. The asymptotic distribution of the maximum size of a tree, in such a forest is studied when $n$, $N\to\infty$ so that $n/N\to 0$, $n/N\to \gamma$ ($0<\gamma<\infty$), $n/N\to\infty$ and $n/N^2\to 0$, or $n/N^2\to\mathrm{const}\ne 0$.