Register context-free grammars (RCFG), register pushdown automata (RPDA) and register tree automata (RTA) are extensions of their classical counterparts to handle data values in a restricted way. These extended models are paid attention as models of query languages for structured documents such as XML with data values. This paper investigates the computational complexity of the basic decision problems for the models. We show that the membership and emptiness problems for RCFG are EXPTIME-complete and also show the membership problem becomes PSPACE-complete and NP-complete for ε-rule free RCFG and growing RCFG, respectively while the emptiness problem remains EXPTIME-complete for these subclasses. The complexities of these problems for RPDA and RTA as well as their language expressive powers are also investigated.