  We study a universal object for the genealogy of a sample in populations with
mutations: the critical birth-death process with Poissonian mutations,
conditioned on its population size at a fixed time horizon. We show how this
process arises as the law of the genealogy of a sample in a large class of
critical branching populations with mutations at birth, namely populations
converging, in a large population asymptotic, towards the continuum random
tree. We extend this model to populations with random foundation times, with
(potentially improper) prior distributions g_i: x\mapsto x^{-i}, i\in\Z_+,
including the so-called uniform (i=0) and log-uniform (i=1) priors.
  We first investigate the mutational patterns arising from these models, by
studying the site frequency spectrum of a sample with fixed size, i.e. the
number of mutations carried by k individuals in the sample. Explicit formulae
for the expected frequency spectrum of a sample are provided, in the cases of a
fixed foundation time, and of a uniform and log-uniform prior on the foundation
time. Second, we establish the convergence in distribution, for large sample
sizes, of the (suitably renormalized) tree spanned by the sample genealogy with
prior g_i on the time of origin. We finally prove that the limiting genealogies
with different priors can all be embedded in the same realization of a given
Poisson point measure.
