In a branching process, suppose $() = + ^2$ for $0 < < 1$ and $ + = 1$. Assume that the population starts with one ancestor. Find $[_3 > 0]$ where $_3$ is the size of the third generation.
For this question, I compute that $P(S)=p+qs^{2}$ follows that $s=p+qs^2$. Therefore the probability of extinction is $s=\frac{1-\sqrt{1-4pq}}{2q}$ and that the mean progeny is $P'(1)=2q$. However, I honestly don't know how to move from this to find $P[Z_3 > 0]$ where $_3$ is the size of the third generation.
Recall that the probability generating function for $Z_n$ is obtained by $n$-fold composition of the probability generating function of the offspring distribution. So \begin{align} P_3(s) &= P(P(P(s)))\\ &= P(P(p+qs^2))\\ &= P(p + q(p+qs^2)^2)\\ &= p+q(p+q(p+qs^2)^2)^2. \end{align} Now, $$ \mathbb P(Z_3=0) = P_3(0) = q \left(p^2 q+p\right)^2+p, $$ so $$ \mathbb P(Z_3>0) = 1-\mathbb P(Z_3=0) = 1-(q \left(p^2 q+p\right)^2+p). $$