  We consider bounded linear operators acting on the $\ell_2$ space indexed by
the nodes of a homogeneous tree. Using the Cuntz relations between the
primitive shifts on the tree, we generalize the notion of the single-scale
time-varying point evaluation and introduce the corresponding reproducing
kernel Hilbert space in which Cauchy's formula holds. These notions are then
used in the study of the Schur multipliers and of the associated de
Branges-Rovnyak spaces. As an application we obtain realization of Schur
multipliers as transfer operators of multiscale input-state-output systems.
