  Consider the generalized iterated wreath product $\mathbb{Z}_{r_1}\wr
\mathbb{Z}_{r_2}\wr \ldots \wr \mathbb{Z}_{r_k}$ where $r_i \in \mathbb{N}$. We
prove that the irreducible representations for this class of groups are indexed
by a certain type of rooted trees. This provides a Bratteli diagram for the
generalized iterated wreath product, a simple recursion formula for the number
of irreducible representations, and a strategy to calculate the dimension of
each irreducible representation. We calculate explicitly fast Fourier
transforms (FFT) for this class of groups, giving literature's fastest FFT
upper bound estimate.
