In this paper, we study deterministic linear pushdown tree automata (deterministic L-PDTAs) and some variations. Since recognition of an input tree by a deterministic L-PDTA can be done in linear time, deterministic L-PDTAs are applicable to many kinds of applications. A strict hierarchy will be shown among the classes of tree languages defined by a variety of deterministic L-PDTAs. It will be also shown that deterministic L-PDTAs are weakly equivalent to nondeterministic L-PDTAs.