Here the definition of wikipedia of Branching process. Let $z_n$ the size of the generation $n$. So, $$Z_{n+1}=\sum_{k=1}^{Z_n}X_{n,i}$$ where $X_{n,i}$ is the number of offspring of the $i-$th individual. We suppose that $Z_0=1$.
My problem is $Z_0=1$. Indeed, shouldn't it be $2$ ? Since we need a man and a woman to make a child ?
The branching process doesn't really model populations undergoing sexual reproduction (as opposed to asexual reproduction). For example, the wikipedia page mentions bacterial reproduction which is asexual (has only one parent) and the historical problem of the genealogy of surnames (where one makes the assumption that surname genealogy is patrilineal so that you only need to consider the male population).
Even if you were interested in a population undergoing sexual reproduction, setting $Z_0 = 2$ wouldn't help since even then reproduction in the model doesn't behave as sexual reproduction. In fact, the process is then the same as the sum of two independent copies of the branching process started from $Z_0 = 1$.