Offspring generating function in Branching process.

Question

Find the extinction probability for a branching process with offspring distribution $a =(1∕6, 1∕2, 1∕3)$.

Solution

The mean of the offspring distribution is
$\mu = 0(1∕6)+ 1(1∕2)+ 2(1∕3)= 7∕6 > 1$,
so this is the supercritical case.

The offspring generating function is $G(s)= 1/6 + s/2 + s^2/3$.

Solving $s = G(s)= 1/6 + s/2 + s^2/3$ gives the quadratic equation $s^2∕3 − s∕2 + 1∕6 = 0$, with roots $s = 1$ and $s = 1∕2$.

The smallest positive root is the probability of eventual extinction $e = 1∕2$.

I have several questions.

1. What would be the offspring generating function in critical and sub-critical cases?
2. Why is $s = G(s)$?