A gene genealogy describes relationships among haplotypes sampled from a population. Knowledge of the gene genealogy for a set of haplotypes is useful for estimation of population genetic parameters and it also has potential application in finding disease-predisposing genetic variants. As the true gene genealogy is unknown, Markov chain Monte Carlo (MCMC) approaches have been used to sample gen...

Consider a continuous-time binary branching process conditioned to have population size n at some time t, and with a chance p for recording each extinct individual in the process. Within the family tree of this process, we consider the smallest subtree containing the genealogy of the extant individuals together with the genealogy of the recorded extinct individuals. We introduce a novel represe...

A well-established model for the genealogy of a large population in equilibrium is Kingman’s coalescent. For the population together with its genealogy evolving in time, this gives rise to a time-stationary tree-valued process. We study the sum of the branch lengths, briefly denoted as tree length, and prove that the (suitably compensated) sequence of tree length processes converges, as the pop...