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

Skeletons of near-critical Bienaym\'{e}-Galton-Watson branching processes