4. If (Zn : 0 ≤ n < ∞) is a branching process in which Z0 = 1 and the size of the r th generation Zr has the generating function Pr (s), prove that Pn(s) = Pr (Pn−r (s)) for 1 ≤ r ≤ n − 1.

Suppose that the process is modified so that the initial generation Z0 is Poisson with parameter

λ, and the number of offspring of each individual is also Poisson with parameter λ. Find a function f such that if Hn(s) is the generating function of the total number of individuals Z0 + Z1 + · · · + Zn, then Hn(s) = f (sHn−1(s)).

(Oxford 1977F)