  We study a linear-fractional Bienaym\'e-Galton-Watson process with a general
type space. The corresponding tree contour process is described by an
alternating random walk with the downward jumps having a geometric
distribution. This leads to the linear-fractional distribution formula for an
arbitrary observation time, which allows us to establish transparent limit
theorems for the subcritical, critical and supercritical cases. Our results
extend recent findings for the linear-fractional branching processes with
countably many types.
