The degree of nondeterminism is a measure of syntactic complexity which was investigated for parallel and sequential rewriting systems. In this paper, we consider the degree of nondeterminsm for tree adjoining grammars and their languages and head grammars and their languages. We show that a degree of nondeterminism of 2 suffices for both formalisms in order to generate all languages in their respective language families. Furthermore, we show that deterministic tree adjoining grammars (those with degree of nondeterminism equal to 1), can generate non-context-free languages, in contrast to deterministic head grammars which can only generate languages containing a single word.