Distributions of cherries and pitchforks for the Ford model

Gursharn Kaur, Kwok Pui Choi and Taoyang Wu

Theoretical Population Biology, 2023, vol. 149, issue C, 27-38

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.

Keywords: Tree shape; Ford’s alpha model; Polya urn model; Random trees; Subtree statistics (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S004058092200079X
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:thpobi:v:149:y:2023:i:c:p:27-38

DOI: 10.1016/j.tpb.2022.12.002

Access Statistics for this article

Theoretical Population Biology is currently edited by Jeremy Van Cleve

More articles in Theoretical Population Biology from Elsevier
Bibliographic data for series maintained by Catherine Liu ().