Limit distribution of the number of trees of given size in a random forest

We consider random forests of a rather general structure. For such forests consisting of $N$ rooted trees and $n$ non-root vertices, as $N,n\to\infty$, we obtain limit distributions of the number of trees of a given size. The special cases of the theorems given are the known results on the forests with labelled vertices and the corresponding results for the forests consisting of plane planted trees and for the forests with constraints on the multiplicities of the vertices.
The work was supported by the Russian Foundation of Basic Research, Grant 94–01–00036–a.