  We study bounded weighted shifts on directed trees. We show that the set of
multiplication operators associated with an injective weighted shift on a
rooted directed tree coincides with the WOT/SOT closure of the set of
polynomials of the weighted shift. From this fact we deduce reflexivity of
those weighted shifts on rooted directed trees whose all path-induced
spectral-like radii are positive. We show that weighted shifts with positive
weights on rooted directed trees admit a Wold-type decomposition. We prove that
the pairwise orthogonality of the factors in the decomposition is equivalent to
the weighted shift being balanced.
