We study the size of the automorphism group of two different types of random trees: Galton-Watson trees and Pólya trees. In both cases, we prove that it asymptotically follows a log-normal distribution. While the proof for Galton-Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of Pólya trees.
@InProceedings{olsson_et_al:LIPIcs.AofA.2022.16, author = {Olsson, Christoffer and Wagner, Stephan}, title = {{Automorphisms of Random Trees}}, booktitle = {33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)}, pages = {16:1--16:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-230-3}, ISSN = {1868-8969}, year = {2022}, volume = {225}, editor = {Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2022.16}, URN = {urn:nbn:de:0030-drops-161026}, doi = {10.4230/LIPIcs.AofA.2022.16}, annote = {Keywords: random tree, Galton-Watson tree, P\'{o}lya tree, automorphism group, central limit theorem} }
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