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Documents authored by Rosenberg, Noah A.

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DOI: 10.4230/LIPIcs.AofA.2024.18
Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves

Authors: Michael R. Doboli, Hsien-Kuei Hwang, and Noah A. Rosenberg

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
The Colijn-Plazzotta ranking is a certain bijection between the unlabeled binary rooted trees and the positive integers, such that the integer associated with a tree is determined from the integers associated with the two immediate subtrees of its root. Letting a_n denote the minimal Colijn-Plazzotta rank among all trees with a specified number of leaves n, the sequence {a_n} begins 1, 2, 3, 4, 6, 7, 10, 11, 20, 22, 28, 29, 53, 56, 66, 67 (OEIS A354970). Here we show that a_n ∼ 2 [2^{P(log₂ n)}]ⁿ, where P varies as a periodic function dependent on {log₂ n} and satisfies 1.24602 < 2^{P(log₂ n)} < 1.33429.
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Michael R. Doboli, Hsien-Kuei Hwang, and Noah A. Rosenberg. Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves. 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. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{doboli_et_al:LIPIcs.AofA.2024.18,
  author =	{Doboli, Michael R. and Hwang, Hsien-Kuei and Rosenberg, Noah A.},
  title =	{{Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{18:1--18:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.18},
  URN =		{urn:nbn:de:0030-drops-204530},
  doi =		{10.4230/LIPIcs.AofA.2024.18},
  annote =	{Keywords: Colijn-Plazzotta ranking, recurrences, unlabeled trees}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.27
Asymptotic Enumeration of Rooted Binary Unlabeled Galled Trees with a Fixed Number of Galls

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

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

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)

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@InProceedings{agranattamir_et_al:LIPIcs.AofA.2024.27,
  author =	{Agranat-Tamir, Lily and Fuchs, Michael and Gittenberger, Bernhard and Rosenberg, Noah A.},
  title =	{{Asymptotic Enumeration of Rooted Binary Unlabeled Galled Trees with a Fixed Number of Galls}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{27:1--27:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.27},
  URN =		{urn:nbn:de:0030-drops-204626},
  doi =		{10.4230/LIPIcs.AofA.2024.27},
  annote =	{Keywords: galled trees, generating functions, phylogenetics, unlabeled trees}
}
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