Sizes of Trees in a Random Forest and Configuration Graphs

We consider Galton–Watson random forests with $N$ rooted trees and $n$ nonroot vertices. The distribution of the number of offspring of the critical homogeneous branching process generating a forest has infinite variance. Such branching processes are used in the study of the structure of random configuration graphs designed for simulating complex communication networks. We prove theorems on the limit distributions of the number of trees of a given size for various relations between $N$ and $n$ as they tend to infinity.