I am trying to understand the derivation of following PGF Formula in branching process
$$H_n(s)=H_{n-1}(H(s))$$
I presume that the initial steps for this derivation are standard...so am avoiding entering them.
The derivation is clear up to the following step
$$H_n(s) = \sum_{i}\Pr(Z_{n-1}=i)\left(\sum_k \Pr(Y_1+Y_2+ \cdots + Y_k=k)s^k\right)$$
The next statement is
Since $\{Y_1, Y_2, \ldots\}$ are iid random variables and PGF of $Y_i$ is $H(S)$
PGF of $Y_1+Y_2+\cdots+Y_i$ is $(H(s))^i$ and hence $H_n(s) = \sum_{i} \Pr(Z_{n-1} = i)(H(s))^i$
I am just unable to fathom this last step..may I request a simple explanation for this last step please?