It is well known that a supercritical single-type Bienayme-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienayme-Galton-Watson processes with countably many types. We find explicit expressions for the main characteristics of the reproduction laws for so-called skeleton and doomed particles.
 F. Klebaner, U. Rosler and S. Sagitov, “Transformations of Galton-Watson Processes and Linear Fractional Reproduction,” Advances in Applied Probability, Vol. 39, No. 4, 2007, pp. 1036-1053. doi:10.1239/aap/1198177238
 S. Sagitov, “Linear-Fractional Branching Processes with Countably Many Types,” 2012, 24 p. http://arxiv.org/abs/1111.4689
 P. Jagers and A. N. Lager?s, “General Branching Processes Conditioned on Extinction Are Still Branching Processes,” Electronic Communications in Probability, Vol. 13, 2008, pp. 540-547. doi:10.1214/ECP.v13-1419