= Solution
If the initial individual has no offspring, $N=0$. If it splits, count that one event and the splitting events in its two <independent> descendant trees. Thus
$$
N=
\begin{cases}
0,&\text{with probability }q,\\
1+N_1+N_2,&\text{with probability }p,
\end{cases}
$$
where $N_1,N_2$ are <independent> copies of $N$. Taking the <probability generating function> gives
$$
G_N(s)=q+psG_N(s)^2.
$$
For $0<s\le1$, the quadratic equation has two algebraic branches. A <probability generating function> must remain finite at $s=0$, where it equals $\mathbb P(N=0)=q$, so the admissible branch is
$$
\boxed{G_N(s)=\frac{1-\sqrt{1-4pqs}}{2ps}
=\frac{2q}{1+\sqrt{1-4pqs}},\qquad G_N(0)=q.}
$$
The rationalized formula is also defined directly at zero. At $s=1$, $\sqrt{1-4pq}=q-p$, so $G_N(1)=1$, consistent with almost-sure finiteness. Extinction occurs after a finite number of generations almost surely, and each generation is finite, which justifies that finiteness claim for the <number of splitting events in a binary branching process>.
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