= Number of splitting events in a binary branching process
{title2=$G(s)=q+psG(s)^2,\qquad \mathbb E[N]=p/(1-2p)$}
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 $G=q+psG^2$ and
$$
G(s)=\frac{2q}{1+\sqrt{1-4pqs}}.
$$
The 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.
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