# Given probabilities for a branching process, how do I compute the probability mass function at a time?

I have a branching process of the form $p_0=0.1$, $p_1 = 0.6$, $p_2 = 0.3$. (any other $p_n = 0$). $Z_0$, the original population is $1$. $Z_1$ is the population after $1$ timestep, $Z_2$ is the population after $2$ timesteps, etc.

How can I compute the probability mass function of $Z_3$? It would suffice to have an expression I could expand on my TI-89.

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What are $p_0$, $p_1$, and $p_2$ the probabilities of? – Rahul Nov 4 '10 at 1:50
Probabilities of generating 0, 1, 2 children respectively. – user2617 Nov 4 '10 at 1:53
Why do you have to use a TI-89? – Yuval Filmus Nov 4 '10 at 1:57

The probability of having $k$ individuals after 3 steps is the coefficient of $x^k$ in the expansion of $f(f(f(x)))$, where $f(x) = 0.1 + 0.6x + 0.3x^2$.