Our concern is the overhead of answering OWL 2QL ontology-mediated queries (OMQs) in ontology-based data access compared to evaluating their underlying tree-shaped and, more generally, bounded treewidth conjunctive queries (CQs). We show that OMQs with bounded depth ontologies have nonrecursive datalog (NDL) rewritings that can be constructed and evaluated in LOGCFL for combined complexity, and even in NL if their CQs are tree-shaped with a bounded number of leaves. Thus, such OMQs incur no overhead in complexity-theoretic terms. For OMQs with arbitrary ontologies and bounded-leaf tree-shaped CQs, NDL-rewritings are constructed and evaluated in LOGCFL. We experimentally demonstrate feasibility and scalability of our rewritings compared to previously proposed NDL-rewritings. On the negative side, we prove that answering OMQs with tree-shaped CQs is not fixed-parameter tractable if the ontology depth or the number of leaves in the CQs is regarded as the parameter, and that answering OMQs with a fixed ontology (of infinite depth) is NP-complete for tree-shaped CQs and LOGCFL-complete for bounded-leaf CQs.