Let $1 = Z_0,Z_1,Z_2,\ldots$ be a Galton-Watson branching process with offspring distribution $p_0,p_1,p_2,\ldots$. That is, $p_k$ is the probability that an individual will have $k$ offspring. Suppose that $p_0 = 2/3$ and $p_2 = 1/3$. Let $V = Z_0 + Z_1 + Z_2 +\cdots$.
Why is it that $P(V < \infty) = 1$, and what is the probability generating function of $V$?
I note that we can try to use a transition matrix, if it helps?