Hi, The article “the” applies to the entire noun phrase, not just to “children’s” and not just to “books”. In “the children’s books”, the head noun is “books”, while “children’s” functions as a possessive modifier. From a grammatical perspective, “the” determines the whole phrase “the children’s books”. It means that the speaker is referring to a specific group of children’s books. For example, “The children’s books are on the shelf” suggests that both the speaker and listener know which books are being discussed. If you want to talk about children’s books in general, you normally leave out the article. For example, “Children’s books are often illustrated.” You could also say, “I enjoy reading children’s books.” In those sentences, the books are not specific and have not been identified previously. So, “the children’s books” refers to a particular set of books, while “children’s books” refers to the category in general.

Here are the children's books you were looking for. This could mean the books belonging to some specific children, or generally. The first could belong to their own children or some other children, the second could mean the age range they were trying to find books for.

The aritcle is for children's. They are [the children]'s books. Because "the" is locked into the possessive phrase "the children," the main noun (books) is technically indefinite and general in this specific structure.