  For branching processes, the generating functions for limit distributions of
so-called ratios of probabilities of rare events satisfy the Schr\"oder-type
integral-functional equations. Excepting limited special cases, the
corresponding equations can not be solved analytically. I found a large class
of Poisson-type offspring distributions, for which the Schr\"oder-type
functional equations can be solved analytically. Moreover, for the asymptotics
of limit distributions, the power and constant factor can be written
explicitly. As a bonus, Bernoulli branching processes in random environments
are treated. The beauty of this example is that the explicit formula for the
generating function is unknown. Still, the closed-form expression for the power
and constant factor in the asymptotic can be written with the help of some
outstanding tricks. Also, the magic power $3$ in the Bernoulli binomial
kernels, at which short-phase discrete oscillations turn into long-phase
continuous ones.
