On conditions for emergence of a giant tree in a random unlabelled forest

We consider the set of random forests consisting of $N$ rooted trees ordered in one of $N!$ possible ways and of $n$ nonroot unlabelled vertices. As $N,n\to\infty$, we find the limit distributions of the $(N-p)$th term of the set of order statistics obtained by arranging the sizes of the trees of a random unlabelled forest in nondescending order for fixed $p=1,2,\dots$ . We find that a giant tree (that is, a tree of size $n+o(n)$) emerges in the only case where $N,n\to\infty$ so that $N/\sqrt n\to0$.