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Distributions of cherries and pitchforks for the Ford model

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  • Kaur, Gursharn
  • Choi, Kwok Pui
  • Wu, Taoyang

Abstract

Distributional properties of tree shape statistics under random phylogenetic tree models play an important role in investigating the evolutionary forces underlying the observed phylogenies. In this paper, we study two subtree counting statistics, the number of cherries and that of pitchforks for the Ford model, the alpha model introduced by Daniel Ford. It is a one-parameter family of random phylogenetic tree models which includes the proportional to distinguishable arrangement (PDA) and the Yule models, two tree models commonly used in phylogenetics. Based on a non-uniform version of the extended Pólya urn models in which negative entries are permitted for their replacement matrices, we obtain the strong law of large numbers and the central limit theorem for the joint distribution of these two statistics for the Ford model. Furthermore, we derive a recursive formula for computing the exact joint distribution of these two statistics. This leads to exact formulas for their means and higher order asymptotic expansions of their second moments, which allows us to identify a critical parameter value for the correlation between these two statistics. That is, when the number of tree leaves is sufficiently large, they are negatively correlated for 0≤α≤1/2 and positively correlated for 1/2<α<1.

Suggested Citation

  • Kaur, Gursharn & Choi, Kwok Pui & Wu, Taoyang, 2023. "Distributions of cherries and pitchforks for the Ford model," Theoretical Population Biology, Elsevier, vol. 149(C), pages 27-38.
    • Handle: RePEc:eee:thpobi:v:149:y:2023:i:c:p:27-38
    DOI: 10.1016/j.tpb.2022.12.002
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    References listed on IDEAS

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    1. Arbisser, Ilana M. & Jewett, Ethan M. & Rosenberg, Noah A., 2018. "On the joint distribution of tree height and tree length under the coalescent," Theoretical Population Biology, Elsevier, vol. 122(C), pages 46-56.
    2. Choi, Kwok Pui & Thompson, Ariadne & Wu, Taoyang, 2020. "On cherry and pitchfork distributions of random rooted and unrooted phylogenetic trees," Theoretical Population Biology, Elsevier, vol. 132(C), pages 92-104.
    3. Wu, Taoyang & Choi, Kwok Pui, 2016. "On joint subtree distributions under two evolutionary models," Theoretical Population Biology, Elsevier, vol. 108(C), pages 13-23.
    4. Wirtz, Johannes & Wiehe, Thomas, 2019. "The Evolving Moran Genealogy," Theoretical Population Biology, Elsevier, vol. 130(C), pages 94-105.
    5. Janson, Svante, 2004. "Functional limit theorems for multitype branching processes and generalized Pólya urns," Stochastic Processes and their Applications, Elsevier, vol. 110(2), pages 177-245, April.
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