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Asymptotic Enumeration of Rooted Binary Unlabeled Galled Trees with a Fixed Number of Galls

Authors Lily Agranat-Tamir , Michael Fuchs , Bernhard Gittenberger , Noah A. Rosenberg

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Author Details

Lily Agranat-Tamir
  • Department of Biology, Stanford University, CA, USA
Michael Fuchs
  • Department of Mathematical Sciences, National Chengchi University, Taipei, Taiwan
Bernhard Gittenberger
  • Department of Discrete Mathematics and Geometry, Technische Universität Wien, Austria
Noah A. Rosenberg
  • Department of Biology, Stanford University, CA, USA

Funding

  • Fuchs, Michael: National Science and Technology Council grant NSTC-111-2115-M-004-002-MY2
  • Rosenberg, Noah A.: National Science Foundation grant BCS-2116322

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Lily Agranat-Tamir, Michael Fuchs, Bernhard Gittenberger, and Noah A. Rosenberg. Asymptotic Enumeration of Rooted Binary Unlabeled Galled Trees with a Fixed Number of Galls. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.AofA.2024.27

Abstract

Galled trees appear in problems concerning admixture, horizontal gene transfer, hybridization, and recombination. Building on a recursive enumerative construction, we study the asymptotic behavior of the number of rooted binary unlabeled (normal) galled trees as the number of leaves n increases, maintaining a fixed number of galls g. We find that the exponential growth with n of the number of rooted binary unlabeled normal galled trees with g galls has the same value irrespective of the value of g ≥ 0. The subexponential growth, however, depends on g; it follows c_g n^{2g-3/2}, where c_g is a constant dependent on g. Although for each g, the exponential growth is approximately 2.4833ⁿ, summing across all g, the exponential growth is instead approximated by the much larger 4.8230ⁿ.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Enumeration
Keywords
  • galled trees
  • generating functions
  • phylogenetics
  • unlabeled trees

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