Of + articles + general references

In US English, there is no effective difference. The sentences that use the definite article are still not talking about a truly specific item since the indefinite article preceding the source negates any particularity.

In US usage the ... a ... is probably more frequent than a ... a ..., perhaps to help the listener focus on the part being described, or perhaps simply to avoid repetition.

SUPPLEMENT

Note that the examples given all talk about hypotheticals. They do not refer to a particular branch, tree, roof or car. While the ... a ... seems to be more definite than a ... a ..., the second indefinite article keeps the discussion hypothetical.

If the construction were changed to have only one article, it would have to be indefinite to preserve the hypothetical form.

A car roof is made of many different materials

and

A tree branch will make decent firewood

but not

The car roof is made of many different materials

or

The tree branch will make decent firewood

The latter two describe a particular roof and branch, not the theoretical roof or branch.

There are times that general reference descriptions use the definite article to describe a single item. In texts or narrations a singular example may stand in for the whole class.

The oak tree is a member of the deciduous group of trees.

This is a rather academic style, and in common speech would seem a bit pedantic.

These are not normal articles.

They are used to mark Generic Noun Phrases of two different types,
which are named after the articles that distinguish them.

• The roof of a car is a Definite Generic, which refers to the Prototype of whatever the noun phrase refers to. That is, it refers to one's own abstract, generalized and idealized percept of what (in this case) the roof of a car looks like and behaves. The Holy Grail of car roofs; what springs to mind when the phrase is heard or used; instantiated in English syntax.

• A roof of a car is an Indefinite Generic, which refers to the Definition of whatever the noun phrase refers to. That is, in order to be a roof of a car, there are necessary properties; in this case, one of them is to be made of many different materials.

In effect, they imply the same thing -- pick a car, any car, and you'll find that, if it has a roof, the roof is made of many materials. So they're both grammatical; perhaps the definite generic would be more common as a strategy, in the case of the car roof. In the case of the branch, the fact that there are many branches per tree changes the prototype, but all that does is to erase any differences -- they're both fine.

If you got this question from a test, be aware that whoever made up the test is incompetent.
These are not good test questions for an English class.