## Publication

2018

### The coalescent of a sample from a binary branching process

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\$$.