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Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves

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

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
Keywords
  • Colijn-Plazzotta ranking
  • recurrences
  • unlabeled trees

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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) https://doi.org/10.4230/LIPIcs.AofA.2024.18

Author Details

Michael R. Doboli
  • Department of Mathematics, Stanford University, CA, USA
Hsien-Kuei Hwang
  • Institute of Statistical Science, Academia Sinica, Taipei, Taiwan
Noah A. Rosenberg
  • Department of Biology, Stanford University, CA, USA

Funding

  • Rosenberg, Noah A.: National Institutes of Health grant R01 HG005855.

Acknowledgements

We are grateful to Michael Fuchs and Daniel Krenn for discussions of this work, and to Stephan Wagner for discussions and for comments on a draft of the manuscript.

References

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