Learn English – Is ‘If to do something, we should…’ grammatically correct

The profits show improvement by time

This means, as time moves forward, profits improve.

The sentence above is not specific as to the interval of the time. You can say any of:

The profits show yearly improvement by time.

The profits show improvement yearly by time.

The profits show improvement by time yearly.

but then "by time" is redundant (because both "by time" and "yearly" modify "improvement" and "yearly" is already a time expression) and doesn't need to be in the sentence, unless you are emphasizing that there is an improvement by time rather than some other metric, such as by customer, etc.

If you say:

The yearly profits show improvement by time.

you are saying a set of yearly profits are steadily improving over a number of years. This is not redundant as "yearly" is qualifying "profits", not "improvement."

(My answer have some limitations, because I myself can't answer beyond that, but without going there if you read my answer you will probably get the answer of your question, not too deep though.)

Do you have [Noun Phrase (NP)]?

Now this NP here is -

Here the PP (prepositional phrase) is the complement of the head noun (suggestion). A PP often acts as a complement or postmodifier of a noun phrase.

The prepositional phrase structure is like this -

There has to be a head preposition - as to/about. Without it the phrase structure is incorrect. And the complement to the head preposition is an embedded wh question.

So from it, it's clear that you can't omit as to or about from the sentence, or it will make it ungrammatical.

Without the preposition there is no connection between the head noun and which book I should buy part. A preposition connects them together.

(My answer doesn't address why the complement to the head noun in the NP tree has to be a Prepositional Phrase. The simple answer is I don't know. I couldn't find the reason beyond what I said here in any grammar book I checked, though I haven't checked many books. I think this will be dealt with not in grammar books, but in good linguistic books, and I don't have any. Nor do I have any knowledge in linguistic theory. Someone else might answer it better.)