When writing about the same thing in several sentences, mainly in definitions, I am often confused about when to switch from the indefinite article to the definite article. Which rule can be applied to this?
Consider this example:
Suppose we have a tree T and a vertex v in T. We say that the vertex v is a leaf in the tree T if its degree is 1.
In the first sentence, I would use the indefinite articles, because I only introduce the terms to the reader.
In the second sentence, however, I would prefer to use "the vertex v", since I have already introduced it and it is clear for both me and the reader what we are talking about. Similarly, I would prefer to use "the tree T". On the other hand, I would use "a leaf", because this term has not been mentioned before.
Is that correct? Or maybe I am just overusing the articles?
This is just a minimum example, maybe trying to be a little too expressive. But I am sure there are real-world situations, where such a switch from the indefinite articles to the definite ones must be made. And I am just wondering how to do it properly.