The man refused to send any letters to that place.
This is what I have so far from the Syntax Tree Generator.
What should replace the X there?
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I am not a fan of X-bar syntax, so I'll just offer a couple of remarks which might be relevant.
Your tree does not show that "send any letters to that place" is a constituent, but clearly it is. (1) It can undergo the rule of VP (or V') deletion: "Henry told the man to send them, but he refused to (0=send them)." (2) it can be coordinated with a like constituent: "The man refused to send them or destroy them." (3) It can be replaced by "do so" (see Lakoff and Ross "Criterion for VP constituency").
The X constituent is a complement of "refuse", but it doesn't seem to be nominal -- at least "He refused it" with "it" referring back to "to send them" sounds pretty bad to me, so that makes X some sort of S or verbal constituent. I think McCawley would make it a V'.
In my own theory How can PSG describe the vertical dimension of sentence structure?, which has no bars but distinguishes 4 degrees of obliqueness of phrases, it's an S2, meaning that it cannot contain any constituent of obliqueness less than 2 (which precludes Tense, Subject, sentence -ly adverb like "possibly", performative adverb, vocative, appositive). The grammatical relation "2" is like the same relation of Relational Grammar, and is the phrasal correspondent to a traditional accusative.