We present a functional interpretation of the substitution and adjunction operations of Tree Adjoining Grammars (TAGs). This allows us to encode TAGs within the formalism of Abstract Categorial Grammars (ACGs). In this encoding, abstract terms representing derivation trees are full-fledged objects of the grammar. These terms are mapped onto logical formulas representing the semantic interpretation of natural expressions that TAG can analyze. Because of the reversibility properties of ACGs, this provides a way to parse and generate with the same TAG encoded grammar. We then show some examples of how we can use and extend this representation of derivation trees, in particular to analyze idioms, subordinate clauses, and scope ambiguities. All the examples can be run with the ACG toolkit.