The genealogy 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 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