Branching Process and point of ceasing Let $(Z_{n})_{n \in \mathbb{N}_{0}}$ be a branching process. The following generations are distributed with the PMF $p(0)=p(2)=\frac{1}{2}$. How do I define the distribution of $Z_{3}$ and what is the probability for the extinction of the process? I started with computing $G(z)=\frac{1}{2}+\frac{1}{2}z^{2}$. How do I continue? Thank you for your help!

The generating function of $Z_3$ is $G(G(G(z)))$. Once you compute this function you can write down the distribution of $Z_3$. The extinction probability is obtained by solving the equation $G(z)=z$ which shows that extinction probability is $1$.