From An Intermediate Course in Probability by Allan Gut:
Consider the following modification of a branching process: A mature >individual produces Children according to the generateing function g(t). >However, an individual becomes mature with probability $\alpha$ and dies >Before maturity with probability $1-\alpha$. Troughout $X(0)=1$, that is, we start with one imature individual.
(a) Find the generating function of the number of individuals in the first two >generations.
(b) Suppose the offspring distribution is geometric with parameter p. >Determine the extinction probability.
The solution to (a) is
$g_{X(1)}(t)=1-\alpha + \alpha g(t),\qquad g_{X(2)}(t)=1-\alpha + \alpha g(1-\alpha + \alpha g(t))$.
(b)
$g_{X(1)}(t)=1-\alpha +\alpha(\frac{p}{1-qt})=\frac{(1-\alpha)(1-qt)+\alpha p}{1-qt}$
$g_{X(2)}(t)=\frac{(1-\alpha)(1-q(\frac{(1-\alpha)(1-qt)+\alpha p}{1-qt}))+\alpha p}{1-q(\frac{(1-\alpha)(1-qt)+\alpha p}{1-qt})}=\frac{\frac{(1-\alpha)(1-qt-q((1-\alpha)(1-qt)+ap))+ap(1-qt)}{1-qt}}{\frac{1-qt-q((1-\alpha)(1-qt)+ap)}{1-qt}}=\frac{(1-\alpha)(1-qt-q((1-\alpha)(1-qt)+ap))+ap(1-qt)}{1-qt-q((1-\alpha)(1-qt)+ap)}$
$\vdots$
I could not find a pattern here to find the generating function for the branching process by induction, to find the extinciton probability as a root of the equation $t=g(t)$.
How to find the extinction probability?
The soulution: b)
$\mathrm{E}X=g'(1)=\frac{\alpha q}{p}$
$t=g(t)$
$t=1-\alpha +\alpha\frac{p}{1-qt}$
$t(1-qt)=(1-\alpha)(1-qt)+\alpha (1-q)$
$0=(1-\alpha)(1-qt)-t(1-qt)+\alpha (1-q)$
$0=qt^{2}-t+1-qt-\alpha +\alpha qt+\alpha -\alpha q=qt^{2}+(\alpha q -q-1)t+1-\alpha q$
Since $t=1$ is always a root of $t=g(t)$, we can find the other root by polynomial long division.
$\underline{t-1}|^{^{qt+\alpha q -1}}\overline{qt^{2}+(\alpha q -q-1)t+1-\alpha q}$
$qt^{2}+(\alpha q -q-1)t+1-\alpha q-qt(t-1)=(\alpha q -1)t-(\alpha q -1)$
$(\alpha q -1)t-(\alpha q -1)-(\alpha q -1)(t-1)=0$
$0=(t-1)(qt+\alpha q -1)$
$0=(t-1)(t-\frac{(1-\alpha q)}{q})$
$t=1, \qquad t=\frac{(1-\alpha q)}{q}$
The probability of extinction, $\eta$, is
$\eta=1 \quad \text{for} \quad\frac{\alpha q}{p}\leq 1,\qquad \frac{(1-\alpha q)}{q} \quad \text{for}\quad \frac{\alpha q}{p}> 1$