Register automaton (RA), register context-free grammar (RCFG) and register tree automaton (RTA) are computational models with registers which deal with data values. This paper shows pumping lemmas for the classes of languages expressed by RA, RCFG and RTA. Among them, the first lemma was already proved in terms of nominal automata, which is an abstraction of RA. We define RTA in a deterministic and bottom-up manner. For these languages, the notion of ‘pumped word’ must be relaxed in such a way that a pumped subword is not always the same as the original subword, but is any word equivalent to the original subword in terms of data type defined in this paper. By using the lemmas, we give examples of languages that do not belong to the above-mentioned classes of languages.