modifiers. Some of these adjective phrases are reduced relative clauses, such as balloons full of helium (compare balloons that were full of helium). Others are post-positive adjective phrases, such as the attorney general. Noun phrases themselves can function as post-head adjuncts in noun phrases. In the noun phrase shoes that size, for instance, the noun phrase that size functions as a post-head modifier. Certain determiners (namely, each, enough, less, and more) can head determiner phrases that function as post-head modifiers of noun phrases, as in the determiner phrase each in three dollars each. Rarely, adverb phrases can function as post-head modifiers, such as the adverb phrase soon in the noun phrase some day soon. External modifiers exist inside the NP but outside the nominal. These modifiers are often adverb phrases, as exemplified by even in the tree diagram above. External modifiers can also be realized by prepositional phrases (e.g., by far the greatest ally) and noun phrases (e.g., every bit a philosopher). External modifiers can only attach to the beginnings or ends of noun phrases. When positioned at the beginning, they occur before any predeterminative, determinative, or internal modifier. In the noun phrase even all their best songs, for instance, the external modifier (even) must occur before the predeterminative (all), determinative (their), and internal modifier (best). Some external modifiers can move freely between the beginning and the end of their noun phrase. For example, by far the greatest ally can also be written the greatest ally by far. Complements A nominal can occasionally include a complement, a dependent licensed by the head noun. Usually, these are prepositional phrases or subordinate clauses. The head of such prepositional phrases is typically of, as in our review of your application or your receipt of the envelope. In some of these cases, the complement and noun can be compared to a verb and direct object pair (we reviewed your application; you received the envelope). In other cases, the head is not of, as exemplified by about exercise in the tree diagram above. Clauses that function as complements in noun phrases can be either finite (a realization that it is important) or non-finite (a requirement for them to do it). As with prepositional phrase complements of nouns, certain clause complements of nouns can be compared to verb and complement pairs (they realized that it is important; somebody required them to do it). Nouns can also