We present a language model consisting of a collection of costed bidirectional finite state automata associated with the head words of phrases. The model is suitable for incremental application of lexical associations in a dynamic programming search for optimal dependency tree derivations. We also present a model and algorithm for machine translation involving optimal "tiling" of a dependency tree with entries of a costed bilingual lexicon. Experimental results are reported comparing methods for assigning cost functions to these models. We conclude with a discussion of the adequacy of annotated linguistic strings as representations for machine translation.