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Measuring the external branches of a Kingman tree: A discrete approach

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  • Disanto, Filippo
  • Wiehe, Thomas

Abstract

The Kingman coalescent process is a classical model of gene genealogies in population genetics. It generates Yule-distributed, binary ranked tree topologies – also called histories – with a finite number of n leaves, together with n−1 exponentially distributed time lengths: one for each layer of the history. Using a discrete approach, we study the lengths of the external branches of Yule distributed histories, where the length of an external branch is defined as the rank of its parent node. We study the multiplicity of external branches of given length in a random history of n leaves. A correspondence between the external branches of the ordered histories of size n and the non-peak entries of the permutations of size n−1 provides easy access to the length distributions of the first and second longest external branches in a random Yule history and coalescent tree of size n. The length of the longest external branch is also studied in dependence of root balance of a random tree. As a practical application, we compare the observed and expected number of mutations on the longest external branches in samples from natural populations.

Suggested Citation

  • Disanto, Filippo & Wiehe, Thomas, 2020. "Measuring the external branches of a Kingman tree: A discrete approach," Theoretical Population Biology, Elsevier, vol. 134(C), pages 92-105.
    • Handle: RePEc:eee:thpobi:v:134:y:2020:i:c:p:92-105
    DOI: 10.1016/j.tpb.2020.05.002
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    References listed on IDEAS

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    1. Dhersin, Jean-Stéphane & Freund, Fabian & Siri-Jégousse, Arno & Yuan, Linglong, 2013. "On the length of an external branch in the Beta-coalescent," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1691-1715.
    2. Erik M. Rauch & Yaneer Bar-Yam, 2004. "Theory predicts the uneven distribution of genetic diversity within species," Nature, Nature, vol. 431(7007), pages 449-452, September.
    3. Zhu, Sha & Degnan, James H. & Steel, Mike, 2011. "Clades, clans, and reciprocal monophyly under neutral evolutionary models," Theoretical Population Biology, Elsevier, vol. 79(4), pages 220-227.
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