Machine Translation (MT) systems based on Data-Oriented Parsing (DOP: Bod, 1998) and LFG-DOP (Bod & Kaplan, 1998) may be viewed as instances of Example-Based MT (EBMT). In both approaches, new translations are processed with respect to previously seen translations residing in the system's database. We describe DOT models for translation (Poutsma, 1998, Poutsma 2000) based on DOP. We demonstrate that DOT1 is not guaranteed to produce the correct translation, despite provably deriving the most probable translation. The DOT2 translation model solves most of the problems of DOT1, but suffers from limited compositionality when confronted with certain data. Nothwithstanding the success of DOT2, any system based purely on trees will ultimately be found wanting as a general solution to the wide diversity of translation problems, as certain linguistic phenomena require a description at levels deeper than surface syntax. We then show how LFG-DOP can be extended to serve as a novel hybrid model for MT, LFG-DOT (Way, 2001), which promises to improve upon DOT and other EBMT systems.