Decomposable branching processes with a fixed extinction moment

The asymptotic behavior as n -> a of the probability of the event that a decomposable critical branching process Z(m) = (Z (1)(m),..., Z (N)(m)), m = 0, 1, 2,..., with N types of particles dies at moment n is investigated, and conditional limit theorems are proved that describe the distribution of the number of particles in the process Z(center dot) at moment m < n given that the extinction moment of the process is n. These limit theorems can be considered as statements describing the distribution of the number of vertices in the layers of certain classes of simply generated random trees of fixed height.