  We present an elementary model of random size varying population given by a
stationary continuous state branching process. For this model we compute the
joint distribution of: the time to the most recent common ancestor, the size of
the current population and the size of the population just before the most
recent common ancestor (MRCA). In particular we show a natural mild bottleneck
effect as the size of the population just before the MRCA is stochastically
smaller than the size of the current population. We also compute the number of
old families which corresponds to the number of individuals involved in the
last coalescent event of the genealogical tree. By studying more precisely the
genealogical structure of the population, we get asymptotics for the number of
ancestors just before the current time. We give explicit computations in the
case of the quadratic branching mechanism. In this case, the size of the
population at the MRCA is, in mean, less by 1/3 than size of the current
population size. We also provide in this case the fluctuations for the
renormalized number of ancestors.
