  Skeletons of branching processes are defined as trees of lineages
characterized by an appropriate signature of future reproduction success. In
the supercritical case a natural choice is to look for the lineages that
survive forever. In the critical case it was earlier suggested to distinguish
the particles with the total number of descendants exceeding a certain
threshold. These two definitions lead to asymptotic representations of the
skeletons as either pure birth process (in the slightly supercritical case) or
critical birth-death processes (in the critical case conditioned on the total
number of particles exceeding a high threshold value). The limit skeletons
reveal typical survival scenarios for the underlying branching processes.
  In this paper we consider near-critical Bienaym\'{e}-Galton-Watson processes
and define their skeletons using marking of particles. If marking is rare, such
skeletons are approximated by birth and death processes which can be
subcritical, critical or supercritical. We obtain the limit skeleton for a
sequential mutation model and compute the density distribution function for the
time to escape from extinction.
