We reconsider the problem of containment of monadic data- log (MDL) queries in unions of conjunctive queries (UCQs). Prior work has dealt with special cases, but has left the precise complexity character- ization open. We begin by establishing a 2EXPTIME lower bound on the MDL/UCQ containment problem, resolving an open problem from the early 90's. We then present a general approach for getting tighter bounds on the complexity, based on analysis of the number of mappings of queries into tree-like instances. We use the machinery to present an important case of the MDL/UCQ containment problem that is in co-NEXPTIME, and a case that is in EXPTIME. We then show that the technique can be used to get a new tight upper bound for containment of tree automata in UCQs. We show that the new MDL/UCQ upper bounds are tight.