Elsevier

Discrete Mathematics

Volume 341, Issue 4, April 2018, Pages 896-911
Discrete Mathematics

On the shape of random Pólya structures

https://doi.org/10.1016/j.disc.2017.12.016Get rights and content
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Abstract

Panagiotou and Stufler recently proved an 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 Cn and many small forests, where with probability tending to one, as n tends to infinity, any forest Fn(v), that is attached to a node v in Cn, is maximally of size |Fn(v)|=O(logn). 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 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.

Keywords

Pólya tree
Pólya enumeration theorem
Simply generated tree
Rooted identity tree
Automorphism group of trees

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A preliminary version of this paper was published in the Proceedings of ANALCO 2017.