The coalescent of a sample from a binary branching process

Theor Popul Biol. 2018 Jul:122:30-35. doi: 10.1016/j.tpb.2018.04.005. Epub 2018 Apr 25.


At time 0, start a time-continuous binary branching process, where particles give birth to a single particle independently (at a possibly time-dependent rate) and die independently (at a possibly time-dependent and age-dependent rate). A particular case is the classical birth-death process. Stop this process at time T>0. It is known that the tree spanned by the N tips alive at time T of the tree thus obtained (called a reduced tree or coalescent tree) is a coalescent point process (CPP), which basically means that the depths of interior nodes are independent and identically distributed (iid). Now select each of the N tips independently with probability y (Bernoulli sample). It is known that the tree generated by the selected tips, which we will call the Bernoulli sampled CPP, is again a CPP. Now instead, select exactly k tips uniformly at random among the N tips (a k-sample). We show that the tree generated by the selected tips is a mixture of Bernoulli sampled CPPs with the same parent CPP, over some explicit distribution of the sampling probability y. An immediate consequence is that the genealogy of a k-sample can be obtained by the realization of k random variables, first the random sampling probability Y and then the k-1 node depths which are iid conditional on Y=y.

Keywords: Birth–death process; Coalescent point process; Finite exchangeable sequence.; Incomplete sampling; Random tree; Splitting tree.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Algorithms
  • Animals
  • Binomial Distribution*
  • Birth Rate
  • Death
  • Genealogy and Heraldry
  • Genetics, Population*
  • Humans
  • Models, Genetic*
  • Mortality
  • Parturition
  • Probability*