Reduced subcritical Galton-Watson processes in a random environment

We study the structure of genealogical trees of reduced subcritical Galton-Watson processes in a random environment assuming that all (randomly varying in time) offspring generating functions are fractional linear. We show that this structure may differ significantly from that of the `classicalʹ reduced subcritical Galton-Watson processes. In particular, it may look like a complex `hybridʹ of classical reduced super and subcritical processes. Some relations with random walks in a random environment are discussed.