Although current CCG supertaggers achieve high accuracy on the standard WSJ\ntest set, few systems make use of the categories' internal structure that will\ndrive the syntactic derivation during parsing. The tagset is traditionally\ntruncated, discarding the many rare and complex category types in the long\ntail. However, supertags are themselves trees. Rather than give up on rare\ntags, we investigate constructive models that account for their internal\nstructure, including novel methods for tree-structured prediction. Our best\ntagger is capable of recovering a sizeable fraction of the long-tail supertags\nand even generates CCG categories that have never been seen in training, while\napproximating the prior state of the art in overall tag accuracy with fewer\nparameters. We further investigate how well different approaches generalize to\nout-of-domain evaluation sets.\n