Number of splitting events in a binary branching process ($G(s)=q+psG(s{)}^2,\mathbb{E}[N]=p/(1-2p)$)

Consider a Galton-Watson process started by one individual, with offspring count $2$ with probability $p<1/2$ and $0$ with probability $q=1-p$. Let $N$ count individuals producing two offspring. At the first generation, either $N=0$, or $N=1+N_1+N_2$ for independent copies $N_1,N_2$. Its probability generating function therefore satisfies $G=q+ps{G}^2$ andThe branch is fixed by $G(0)=q$. Differentiation at $s=1$ gives $\mathbb{E}[N]=p/(1-2p)$. Alternatively, generation $t$ has expected size $(2p{)}^t$, so its expected splitting count is $p(2p{)}^t$ and summing gives the same result. The total progeny is $2N+1$, since every split creates two new individuals.