This paper discusses the decidability of node query preservation problems for tree transducers. We assume a transformation given by a deterministic linear top-down data tree transducer (abbreviated as DLTV) and an n-ary query based on runs of a tree automaton. We say that a DLTV Tr strongly preserves a query Q if there is a query Q' such that for every tree t, the answer set of Q' for Tr(t) is equal to the answer set of Q for t. We also say that Tr weakly preserves Q if there is a query Q' such that for every t, the answer set of Q' for Tr(t) includes the answer set of Q for t. We show that the weak preservation problem is coNP-complete and the strong preservation problem is in 2-EXPTIME. We also show that the problems are decidable when a given transducer is a functional extended linear top-down data tree transducer with regular look-ahead, which is a more expressive transducer than DLTV.