 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. Thuswhere $N_1,N_2$ are independent copies of $N$. Taking the probability generating function givesFor $0<s\le 1$, 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 isThe 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.