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Am I analyzing this sentence in syntax correctly?

I don't know how many books you've sold.

S = NP + VP

NP = I

VP = VP(1) + NP(1)

VP(1) = [don't [know]] (don't is a pre-string that modifies the main verb know)

NP(1) = [how many [books] (that) you've sold] (how many is a pre-string and you've sold is a relative clause as a post-string that modify the main noun books)

Is this correct?

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JukerlawVP(1) = [don't [know]] (don't is a pre-string that modifies the main verb know)

I have never encountered that terminology!

The verbs do, have and be are auxiliary verbs, so I would analyze your main clause as

I = subject

do know = verb phrase, consisting of the lexical verb is know, and the auxiliary, do. The auxiliary is required for negation, since lexical verbs are not directly negated in English syntax.

https://www.thoughtco.com/what-is-a-lexical-verb-1691228
https://www.thoughtco.com/nice-properties-verbs-1691346

Not is an adverb.

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I thought every Phrase Type has 3 main parts that are Pre-string - Head - Post-string.

JukerlawI thought every Phrase Type

Prepositional phrases do not have that structure. The head word is the preposition and it comes first, is followed by a complement, most commonly a noun phrase.

https://learnenglish.britishcouncil.org/en/english-grammar/prepositional-phrases


Verb phrases do not have that structure.

Noun phrases can have that structure: premodifiers, head noun, postmodifiers.
Read this: https://learnenglish.britishcouncil.org/en/english-grammar/noun-phrase
Premodifiers can be determiners, quantifiers or adjective phrases.
Postmodifiers can be clauses, prepositional phrases, and (rarely) adjectives and

AlpheccaStars
JukerlawI thought every Phrase Type

Prepositional phrases do not have that structure. The head word is the preposition and it comes first, is followed by a complement, most commonly a noun phrase.

https://learnenglish.britishcouncil.org/en/english-grammar/prepositional-phrases


Verb phrases do not have that structure.

Noun phrases can have that structure: premodifiers, head noun, postmodifiers.
Read this: https://learnenglish.britishcouncil.org/en/english-grammar/noun-phrase
Premodifiers can be determiners, quantifiers or adjective phrases.
Postmodifiers can be clauses, prepositional phrases, and (rarely) adjectives and

Thank you. I'll research more about this.

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