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learnersdictionary.com:
(1a) Algebra, arithmetic, calculus, geometry, and trigonometry are branches of mathematics.
I would like to know what role the presenсe and the absence of "the" before "branches" plays.

my variant:
(1b) Algebra, arithmetic, calculus, geometry, and trigonometry are the branches of mathematics.

What is the difference between (1a) and (1b)?

Also I would like to know whether something will change if I replace "of mathematics" with "mathematics has".

(2a) Algebra, arithmetic, calculus, geometry, and trigonometry are branches mathematics has.
Does the absence of "the" in (2a) have the same meaning as in (1a)?
If not, then why not?

(2b) Algebra, arithmetic, calculus, geometry, and trigonometry are the branches mathematics has.
Does the presence of "the" in (2b) have the same meaning as in (1b)?
If not, then why not?

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  • [The] branches mathematics has is not at all idiomatic. Mathematics isn't a tree with literal branches on it! These 'branches' are sections it can be divided into.
    – Kate Bunting
    Sep 28 at 14:54

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Are the is exhaustive: John, Paul, George, and Ringo are the Beatles means that someone is a Beatle if and only if they are one of the four people listed. By contrast, John and Paul are Beatles asserts something about the two musicians but does not constitute a complete enumeration of all the Beatles.

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    That's the most likely interpretation given the lack of context. You also could use "the branches" to refer to any particular group of branches. Such as: "We offer tuition in three branches of mathematics and two sciences. Algebra, arithmetic, and calculus are the branches of mathematics."
    – Stuart F
    Sep 28 at 15:21
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    @StuartF, “…are the branches of mathematics [in which we offer tuition].”
    – Paul Tanenbaum
    Sep 28 at 16:22

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