  It is well known that a supercritical single-type Bienyam\'e-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 Bienyam\'e-Galton-Watson processes with countably many
types.
