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We present ongoing work on data-driven parsing of German and French with Lexicalized Tree Adjoining Grammars. We use a supertagging approach combined with deep learning. We show the challenges of extracting LTAG supertags from the French Treebank, introduce the use of left- and right-sister-adjunction, present a neural architecture for the supertagger, and report experiments of n-best supertagging for French and German.