Suppose that I have the following branching process. Each parent can have up to two children, the number of which is determined by two independent fair coin flips.
I know that this branching process will eventually go extinct, since each parent runs a Binomial trial with parameters $n=2$, $p=0.5$ and hence $np<1$.
I'm interested in finding a probability bound on the size of the population that this branching process produces. For example, a statement of the form: with probability $q$ the population will be at most $N$ before the process dies out.
I would be great if someone could point me to a reference.
There can be two interpretations of your question. Under the first, you are interested in the maximal population existing at any given time, i.e., $\max_i Y_i$, where $Y_i$ is the population at time $i$. Under the second, you are interested in what is knows as total progeny, which is $\sum_i Y_i$.
Lindvall showed that for a critical branching process, $n\Pr[M>n] \to r$, where $M = \max_i Y_i$ and $r = Y_0$ is the initial population.
Dwass showed that for a subcritical or critical branching process, $\Pr[Z=n] = \frac{r}{n} p_{n-r}^{(n)}$, where $Z = \sum_i Y_i$, $r = Y_0$ is the initial population, and $p_{n-r}^{(n)}$ is the probability that $n$ steps of the process applied to a single parent result in $n-r$ offsprings, i.e. the coefficient of $x^{n-r}$ in $P(x)^n$, where $P(x)$ is the generating function of the process (in your case, $P(x) = (1+x)^2/4$).