On the shape of random Pólya structures

In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for |Fn(v)|, namely |Fn(v)|=Θ(logn). Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Third, we derive the limit probability that for a random node v the attached forest Fn(v) is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other Pólya structures.