A note on the scaling limits of random Pólya trees

Panagiotou and Stufler (arXiv:1502.07180v2) recently proved one important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size n consists of a conditioned critical Galton-Watson tree Tn and many small forests, where with probability tending to one as n tends to infinity, each forest Fn(v) is maximally of size |Fn(v)| = O(log n). Their proof used the framework of a Boltzmann sampler and deviation inequalities. In this paper, first we employ a unified framework in analytic combinatorics to prove this fact with additional improvements on the bound of |Fn(v)|, namely |Fn(v)| = Θ(log n). Second we give a combinatorial interpretation of all weights on the D-forests and C-trees in terms automorphisms associated to a given Pólya tree. Finally, we derive the limit probability that for a random node v the attached forest Fn(v) is of a given size.