Generalized Tree-Adjoining Grammar ABSTRACT This paper continues a program extending results related to the descriptive characterization of the CFLs in terms of definability in the weak monadic second-order theory of trees to the TALs and the entire hierarchy of Weir's Control Languages. Previously, we have shown that the languages in this hierarchy can be characterized by definability in the weak monadic second-order theories of a generalization of trees to increasingly higher dimensions. Here, we explore the effect of admitting models with arbitrary finite branching. In the two-dimensional case we have shown previously that this admits tree sets recognized by infinite, but regular, tree-automata. As these sets correspond, in a very strong sense, to the sets of trees licensed by GPSG grammars we have referred to these automata as Generalized Finite-State Tree Automata and the sets they accept as Generalized Recognizable Sets of trees. In lifting this result to the third and higher dimension, we show that the definable sets of structures are exactly those recognized by $d$-dimensional tree automata which are, in essence, generalized recognizable sets of $(d-1)$-dimensional structures. In three-dimensions, these correspond to Tree-Adjoining Grammars in which the set of initial trees can be any generalized recognizable set of trees---a variation we refer to, by analogy, as Generalized Tree Adjoining Grammars. We show that this permits adoption of GPSG-style accounts of coordination which can potentially provide a conceptually clean account of coordinate constructions that have been problematic for TAG in the past.