# Binomial converging to Poisson in branching process view of $G(n,p)$

I'm trying to understand this claim on page 205 of "The Probabilistic Method" by Alon and Spencer:

Set $$p=c/n$$. A key observation is that $$Z_1 \sim Bin[n-1, c/n]$$ approaches (in $$n$$) the Poisson distribution with mean $$c$$. Further, in a more rough sense, the same holds for $$Z_t$$ as long as $$N_{t-1} \sim o(n)$$.

Here $$Z_t \sim Bin[N_{t-1}, c/n]$$. So if $$N_{t-1} \sim o(n)$$ then wouldn't $$Bin[N_{t-1}, c/n]$$ approach a Poisson distribution with mean $$0$$? What am I missing here?

or equivalently, the number of live and dead vertices is $$o(n)$$.
$$N_{t-1}$$ is not the number of live and dead vertices; it is the number of neutral vertices. Since all vertices are live, dead, or neutral, the number of live and dead vertices is $$n - N_{t-1}$$.
So the condition that should make this observation true is $$N_{t-1} = n - o(n)$$, at which point the Poisson approximation is not surprising any more.