  We consider the problem of estimating the elapsed time since the most recent
common ancestor of a finite random sample drawn from a population which has
evolved through a Bienayme-Galton-Watson branching process. More specifically,
we are interested in the diffusion limit appropriate to a supercritical process
in the near-critical limit evolving over a large number of time steps. Our
approach differs from earlier analyses in that we assume the only known
information is the mean and variance of the number of offspring per parent, the
observed total population size at the time of sampling, and the size of the
sample. We obtain a formula for the probability that a finite random sample of
the population is descended from a single ancestor in the initial population,
and derive a confidence interval for the initial population size in terms of
the final population size and the time since initiating the process. We also
determine a joint likelihood surface from which confidence regions can be
determined for simultaneously estimating two parameters, (1) the population
size at the time of the most recent common ancestor, and (2) the time elapsed
since the existence of the most recent common ancestor.
