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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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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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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.

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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

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

References

  1. L. Agranat-Tamir, S. Mathur, and N. A. Rosenberg. Enumeration of rooted binary unlabeled galled trees. Bulletin of Mathematical Biology, 86:45, 2024. URL: https://doi.org/10.1007/s11538-024-01270-8.
  2. F. Bienvenu, A. Lambert, and M. Steel. Combinatorial and stochastic properties of ranked tree-child networks. Random Structures and Algorithms, 60:653-689, 2022. URL: https://doi.org/10.1002/rsa.21048.
  3. M. Bouvel, P. Gambette, and M. Mansouri. Counting phylogenetic networks of level 1 and 2. Journal of Mathematical Biology, 81:1357-1395, 2020. URL: https://doi.org/10.1007/s00285-020-01543-5.
  4. G. Cardona and L. Zhang. Counting and enumerating tree-child networks and their subclasses. Journal of Computer and System Sciences, 114:84-104, 2020. URL: https://doi.org/10.1016/j.jcss.2020.06.001.
  5. K.-Y. Chang, W.-K. Hon, and S. V. Thankachan. Compact encoding for galled-trees and its applications. In 2018 Data Compression Conference, pages 297-306, Snowbird, UT, 2018. URL: https://doi.org/10.1109/DCC.2018.00038.
  6. Y. S. Chang and M. Fuchs. Counting Phylogenetic Networks with Few Reticulation Vertices: Galled and Reticulation-Visible Networks. Bull. Math. Biol., 86(7):76, 2024. URL: https://doi.org/10.1007/s11538-024-01309-w.
  7. Y.-S. Chang, M. Fuchs, and G.-R. Yu. Galled tree-child networks. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, volume 302, Article 2 of Leibniz International Proceedings in Informatics, LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Wadern, 2024. Google Scholar
  8. L. Comtet. Advanced Combinatorics. Reidel, Boston, 1974. URL: https://doi.org/10.1007/978-94-010-2196-8.
  9. J. Felsenstein. Inferring Phylogenies. Sinauer Associates Inc., Sunderland MA, 2004. Google Scholar
  10. P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, Cambridge, 2009. URL: https://doi.org/10.1017/CBO9780511801655.
  11. M. Fuchs, B. Gittenberger, and M. Mansouri. Counting phylogenetic networks with few reticulation vertices: tree-child and normal networks. Australasian Journal of Combinatorics, 73:385-423, 2019. Google Scholar
  12. M. Fuchs, B. Gittenberger, and M. Mansouri. Counting phylogenetic networks with few reticulation vertices: exact enumeration and corrections. Australasian Journal of Combinatorics, 81:257-282, 2021. Google Scholar
  13. M. Fuchs, E.-Y. Huang, and G.-R. Yu. Counting phylogenetic networks with few reticulation vertices: a second approach. Discrete Applied Mathematics, 320:140-149, 2022. URL: https://doi.org/10.1016/j.dam.2022.03.026.
  14. D. Gusfield. ReCombinatorics. MIT Press, Cambridge MA, 2014. Google Scholar
  15. D. Gusfield, S. Eddhu, and C. Langley. Efficient reconstruction of phylogenetic networks with constrained recombination. In Proceedings of the IEEE Computer Society Conference on Bioinformatics, pages 363-374, 2003. URL: https://doi.org/10.1109/CSB.2003.1227337.
  16. D. Gusfield, S. Eddhu, and C. H. Langley. The fine structure of galls in phylogenetic networks. INFORMS Journal on Computing, 16:459-469, 2004. URL: https://doi.org/10.1287/ijoc.1040.0099.
  17. D. H. Huson, R. Rupp, and C. Scornavacca. Phylogenetic Networks: Concepts, Algorithms and Applications. Cambridge University Press, Cambridge, 2010. URL: https://doi.org/10.1017/CBO9780511974076.
  18. S. Kong, J. C. Pons, L. Kubatko, and K. Wicke. Classes of explicit phylogenetic networks and their biological and mathematical significance. Journal of Mathematical Biology, 84:47, 2022. URL: https://doi.org/10.1007/s00285-022-01746-y.
  19. M. Mansouri. Counting general phylogenetic networks. Australasian Journal of Combinatorics, 83:40-86, 2022. Google Scholar
  20. S. Mathur and N. A. Rosenberg. All galls are divided into three or more parts: recursive enumeration of labeled histories for galled trees. Algorithms in Molecular Biology, 18:1, 2023. URL: https://doi.org/10.1186/s13015-023-00224-4.
  21. C. Semple and M. Steel. Phylogentics. Oxford University Press, Oxford, 2003. Google Scholar
  22. C. Semple and M. Steel. Unicyclic networks: compatibility and enumeration. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 3:84-91, 2006. URL: https://doi.org/10.1109/TCBB.2006.14.
  23. Y. S. Song. A concise necessary and sufficient condition for the existence of a galled-tree. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 3:186-191, 2006. URL: https://doi.org/10.1109/TCBB.2006.15.
  24. T. Warnow. Computational Phylogenetics. Cambridge University Press, Cambridge, 2018. URL: https://doi.org/10.1017/9781316882313.
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