  We study multi-type Bienaym\'e-Galton-Watson processes with linear-fractional
reproduction laws using various analytical tools like contour process, spinal
representation, Perron-Frobenius theorem for countable matrices, renewal
theory. For this special class of branching processes with countably many types
we present a transparent criterion for $R$-positive recurrence with respect to
the type space. This criterion appeals to the Malthusian parameter and the mean
age at childbearing of the associated linear-fractional Crump-Mode-Jagers
process.
