  Coalescence processes have received a lot of attention in the context of
conditional branching processes with fixed population size and non-overlapping
generations. Here we focus on similar problems in the context of the standard
unconditional Bienaym\'e-Galton-Watson branching processes, either
(sub)-critical or supercritical. Using an analytical tool, we derive the
structure of some counting aspects of the ancestral genealogy of such
processes, including: the transition matrix of the ancestral count process and
an integral representation of various coalescence times distributions, such as
the time to most recent common ancestor of a random sample of arbitrary size,
including full size.
  We illustrate our results on two important examples of branching mechanisms
displaying either finite or infinite reproduction mean, their main interest
being to offer a closed form expression for their probability generating
functions at all times. Large time behaviors are investigated.
