I'm trying to solve the following exercise but I can't seem to solve it.
A branching process $(X_n :n \geq 0)$ has $P(X_0 = 1) = 1$. Let the total number of individuals in the first $n$ generations of the process $Z_n$, with probability generating function $Q_n$. Prove that for $n \geq 2$
$$ Q_n(s) = sP_1(Q_{n-1}(s))$$ with $P_1$ is the probability generating function $Q_n$ of the family-size distribution.
What I immediately understood, was that we have to relate $Z_n$, the number of individuals in the first $n$ generations, to $Z_{n-1}$, which is the number of individuals in the first $n-1$ generations. Logically: $Z_n = Z_{n-1} + C$ , where we call $C$ the number of individuals in the $n$'th generation. Note that $Z_{n-1}$ and $C$ are independent.
$$Q_n(s) = E(s^{Z_n}) = E(s^{Z_{n-1}+C}) = E(s^{Z_{n-1}})E(C) = Q_{n-1}(s)E(C)$$.
I believe this is not what I should be doing, does anyone have any hints on how to attack this problem?
Thanks for your time.