Automorphisms of Random Trees

Authors Christoffer Olsson , Stephan Wagner

Thumbnail PDF


  • Filesize: 0.64 MB
  • 16 pages

Document Identifiers

Author Details

Christoffer Olsson
  • Department of Mathematics, Uppsala University, Sweden
Stephan Wagner
  • Department of Mathematics, Uppsala University, Sweden

Cite AsGet BibTex

Christoffer Olsson and Stephan Wagner. Automorphisms of Random Trees. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Generating functions
  • random tree
  • Galton-Watson tree
  • Pólya tree
  • automorphism group
  • central limit theorem


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. László Babai. Automorphism groups, isomorphism, reconstruction. In Handbook of combinatorics, Vol. 1, 2, pages 1447-1540. Elsevier Sci. B. V., Amsterdam, 1995. Google Scholar
  2. Miklós Bóna and Philippe Flajolet. Isomorphism and symmetries in random phylogenetic trees. J. Appl. Probab., 46(4):1005-1019, 2009. URL:
  3. Michael Drmota. Random trees. SpringerWienNewYork, Vienna, 2009. An interplay between combinatorics and probability. URL:
  4. Cecilia Holmgren and Svante Janson. Limit laws for functions of fringe trees for binary search trees and random recursive trees. Electron. J. Probab., 20:no. 4, 51, 2015. URL:
  5. Cecilia Holmgren, Svante Janson, and Matas Šileikis. Multivariate normal limit laws for the numbers of fringe subtrees in m-ary search trees and preferential attachment trees. Electron. J. Combin., 24(2):Paper No. 2.51, 49, 2017. URL:
  6. Mikhail Isaev, Angus Southwell, and Maksim Zhukovskii. Distribution of tree parameters by martingale approach, 2019. URL:
  7. Svante Janson. Random cutting and records in deterministic and random trees. Random Structures & Algorithms, 29(2):139-179, 2006. URL:
  8. Svante Janson. Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton-Watson trees. Random Structures Algorithms, 48(1):57-101, 2016. URL:
  9. Olav Kallenberg. Foundations of modern probability. Springer, New York, 1997. Google Scholar
  10. David Matthews. Automorphisms of Random Recursive Trees. PhD thesis, University of Southhampton, 2017. URL:
  11. Kathleen A. McKeon. The expected number of symmetries in locally-restricted trees. I. In Graph theory, combinatorics, and applications. Vol. 2 (Kalamazoo, MI, 1988), Wiley-Intersci. Publ., pages 849-860. Wiley, New York, 1991. Google Scholar
  12. Kathleen A. McKeon. The expected number of symmetries in locally restricted trees. II. Discrete Appl. Math., 66(3):245-253, 1996. URL:
  13. Dimbinaina Ralaivaosaona and Stephan Wagner. A central limit theorem for additive functionals of increasing trees. Combin. Probab. Comput., 28(4):618-637, 2019. URL:
  14. Dimbinaina Ralaivaosaona, Matas Šileikis, and Stephan Wagner. A central limit theorem for almost local additive tree functionals. Algorithmica, 82(3):642-679, 2020. URL:
  15. Stephan Wagner. Central limit theorems for additive tree parameters with small toll functions. Combinatorics, Probability and Computing, 24(1):329-353, 2015. URL:
  16. Le Yu. Automorphisms of random trees. PhD thesis, Drexel University, 2012. URL: