Limit distribution of given size trees’ number in Galton-Watson forest with limited number of vertices

A subcritical or a critical homogeneous Galton - Watson process starting with N particles is considered. In this process, the number of offsprings from every particle has a Poisson distribution. The set of realizations of such process is a set of forests consisting of N rooted trees with labeled vertices, and the probability distribution on this set is naturally induced by the branching process. Such random forests are known as Galton - Watson forests. Under N,n-œ we obtained limit distributions of the number of trees of a given size for a subset of Galton - Watson forests, in which the total number of vertices does not exceed n.