Principal abstract families of weighted tree languages

We introduce semiring-weighted regular tree grammars with storage where the semiring is complete. We show that the class of weighted tree languages generated by them is a principal abstract family of weighted tree languages provided that the semiring is commutative, the rank of symbols occurring in trees is bounded by a global parameter, and that the storage is finitely encoded and contains a reset instruction. Moreover, we prove the same statement for the iterated pushdown (instead of finitely encoded storage containing a reset instruction).