 In the sentence, they exist quite independently of other people, we have an adverb phrase. To understand its internal structure, let's build it step by step. Clearly, QUITE is a degree adverb which modifies the adverb independently. So the head is the adverb and we get an adverb phrase. But what happens if we add the prepositional phrase of other people, whose internal structure doesn't matter here and can thus be presented as a triangle? Which node is the prepositional phrase's mother node? Clearly, the prepositional phrase must form a constituent with independently. If we apply the proform test, we see that we can replace the whole construction independently of other people by so. In other words, we have to redraw the tree. Here is the categorical analysis again and now the expanded phrasal analysis. As a result, we now have two adverb phrases, the highest one with a degree adverb as a modifier and the internal one with a head and an optional extension. And as suggested by Chomsky in 1970, we refer to the internal one as adverb bar, which today is more conveniently spelled as adverb with an apostrophe.