Expectation of a Galton-Watson process

Question

The base hypothesis are these:

Let {$X(n), n ≥ 0$} denote the number of individuals in generation $n$ in a Galton-Watson process ($X(0) = 1$), and suppose that the mean off-spring is finite and equal to $m$.

I need to show that $\mathbb{E}X(n) = m^n$

Answer 1

Galton-Watson process is a branching process. For a branching process $X_{n}= Y_{1} +\dots + Y_{X_{n-1}}$ we have: $$ E[X_{n}] = E[Y]E[X_{n-1}] = mE[X_{n-1}]. $$

Then, $E[X_{n}] = m^{n}$.