  A rotated odometer is an infinite interval exchange transformation (IET)
obtained as a composition of the von Neumann-Kakutani map and a finite IET of
intervals of equal length. In this paper, we consider rotated odometers for
which the finite IET is of intervals of length $2^{-N}$, for some $N \geq 1$.
We show that every such system is measurably isomorphic to a
$\mathbb{Z}$-action on a rooted tree, and that the unique minimal aperiodic
subsystem of this action is always measurably isomorphic to the action of the
adding machine. We discuss the applications of this work to the study of group
actions on binary trees.
