  We present a generalization of bilateral weighted shift operators for the
noncommutative multivariable setting. We discover a notion of periodicity for
these shifts, which has an appealing diagramatic interpretation in terms of an
infinite tree structure associated with the underlying Hilbert space. These
shifts arise naturally through weighted versions of certain representations of
the Cuntz C$^*$-algebras $O_n$. It is convenient, and equivalent, to consider
the weak operator topology closed algebras generated by these operators when
investigating their joint reducing subspace structure. We prove these algebras
have non-trivial reducing subspaces exactly when the shifts are
doubly-periodic; that is, the weights for the shift have periodic behaviour,
and the corresponding representation of $O_n$ has a certain spatial
periodicity. This generalizes Nikolskii's Theorem for the single variable case.
