  We introduce weighted regular tree grammars with storage as combination of
(a) regular tree grammars with storage and (b) weighted tree automata over
multioperator monoids. Each weighted regular tree grammar with storage
generates a weighted tree language, which is a mapping from the set of trees to
the multioperator monoid. We prove that, for multioperator monoids canonically
associated to particular strong bi-monoids, the support of the generated
weighted tree languages can be generated by (unweighted) regular tree grammars
with storage. We characterize the class of all generated weighted tree
languages by the composition of three basic concepts. Moreover, we prove
results on the elimination of chain rules and of finite storage types, and we
characterize weighted regular tree grammars with storage by a new weighted
MSO-logic.
