The genealogy of a sample from a binary branching process

Abstract: 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 reduced tree or coalescent tree) is a coalescent point process (CPP), which basically means that the depths of interior nodes are 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 Bernoulli sampled CPP, is again a CPP. Now instead, select exactly $k$ tips uniformly at random among the $N$ tips ($k$-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 $Y=y$.