Limit distributions for the maximum size of a tree in a random forest

We consider random forests consisting of $N$ rooted trees with $n$ non-root vertices. The constraints imposed on the structure of trees of such forests are of the quite general nature; forests with labelled vertices, in particular, with constraints on the degrees of vertices, as well as forests of plane planted trees, satisfy them. We obtain the limit distributions of the maximum size of trees for various domains of variations of $N$ and $n$.