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On the joint distribution of tree height and tree length under the coalescent

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  • Arbisser, Ilana M.
  • Jewett, Ethan M.
  • Rosenberg, Noah A.

Abstract

Many statistics that examine genetic variation depend on the underlying shapes of genealogical trees. Under the coalescent model, we investigate the joint distribution of two quantities that describe genealogical tree shape: tree height and tree length. We derive a recursive formula for their exact joint distribution under a demographic model of a constant-sized population. We obtain approximations for the mean and variance of the ratio of tree height to tree length, using them to show that this ratio converges in probability to 0 as the sample size increases. We find that as the sample size increases, the correlation coefficient for tree height and length approaches (π2−6)∕[π2π2−18]≈0.9340. Using simulations, we examine the joint distribution of height and length under demographic models with population growth and population subdivision. We interpret the joint distribution in relation to problems of interest in data analysis, including inference of the time to the most recent common ancestor. The results assist in understanding the influences of demographic histories on two fundamental features of tree shape.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:thpobi:v:122:y:2018:i:c:p:46-56
    DOI: 10.1016/j.tpb.2017.10.008
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    References listed on IDEAS

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    1. Drmota, Michael & Iksanov, Alex & Moehle, Martin & Roesler, Uwe, 2007. "Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1404-1421, October.
    2. Miroshnikov, Alexey & Steinrücken, Matthias, 2017. "Computing the joint distribution of the total tree length across loci in populations with variable size," Theoretical Population Biology, Elsevier, vol. 118(C), pages 1-19.
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    Cited by:

    1. 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.
    2. 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.
    3. Alimpiev, Egor & Rosenberg, Noah A., 2022. "A compendium of covariances and correlation coefficients of coalescent tree properties," Theoretical Population Biology, Elsevier, vol. 143(C), pages 1-13.

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