Tree Regular Model Checking (TRMC) is the name of a family of techniques for analyzing infinite-state systems in which states are represented by terms, and sets of states by Tree Automata (TA). The central problem in TRMC is to decide whether a set of bad states is reachable. The problem of computing a TA representing (an over- approximation of) the set of reachable states is undecidable, but efficient solutions based on completion or iteration of tree transducers exist. Unfortunately, the TRMC framework is unable to efficiently capture both the complex structure of a system and of some of its features. As an example, for JAVA programs, the structure of a term is mainly exploited to capture the structure of a state of the system. On the counter part, integers of the java programs have to be encoded with Peano numbers, which means that any algebraic operation is potentially represented by thousands of applications of rewriting rules. In this paper, we propose Lattice Tree Automata (LTAs), an extended version of tree automata whose leaves are equipped with lattices. LTAs allow us to represent possibly infinite sets of interpreted terms. Such terms are capable to represent complex domains and related operations in an efficient manner. We also extend classical Boolean operations to LTAs. Finally, as a major contribution, we introduce a new completion-based algorithm for computing the possibly infinite set of reachable interpreted terms in a finite amount of time.