Alexg has got it right, in my view. However, since OP says he is waiting for someone to provide a generalized answer, here's mine.
It is hardly ever wrong to omit the article. "The Mall" is the name on the signs, so must be used: "Strand" (the formal name) is both awkward and confusing, so 'the Strand' is usual: most English towns have a few similar names.
Otherwise, there are many names that have developed from descriptions; 'London Road' is the classic example. Most towns in the Home Counties have a road that leads towards London, and refer to it as 'the London road'. Often, when street names were being given, it was named "London Road". In such a case, locals will often call it 'the London Road', while outsiders including the Post Office call it 'London Road'; I wouldn't say either was right or wrong. (Road is, in practice, the only term to which this applies: "the High Street" is usual, but so is "Church Lane is the high street in that village.")
Similar rules apply to stations, airports, roundabouts, etc. Bournemouth has a roundabout with a Frizzell office block, which everyone calls "the Frizzell roundabout". The council put up a sign saying "Frizzell Roundabout", so you can call it either. As far as I can see, all names with articles follow this rule: you can call what used to be Eastleigh Airport (the airport for Southampton) either "Southampton Airport" or "the Southampton airport". "The Southampton Airport" is not correct, but is an understandable mistake; if enough people use it, the name will change.
One last purely national point; in theory, you could refer to a railway terminus named 'Thingtown Central' as either "Central Station" or "the Central"; maybe this happens in the US. Britain has too many places like Exeter, where Exeter Central is a suburban halt, and the central station is Exeter St David's. (The explanation is historical.) So "the Central Station" would be highly ambiguous, and is never used.
I think your description of why you used each article is very close.
In general, we use "the" when the thing is the only one, or we are referring to a specific one. We use "a" when it is an indefinite one of many. Whether there is one or many depends on context: When we use "the", we are not necessarily saying that this is the only one in the history of the universe, but the only one relevant in the present context.
For example: (a) "I entered a room full of strangers. A man named John approached me." There could be many men named "John" there, so I should use the indefinite article. (b) "I met three men who said their names were Bob, Paul, and John. The man named John approached me." Now in context there is only one relevant man named "John", so I should use the definite article.
In your example, the first sentence introduces the tree and the vertex. So at that instant, they are indefinite. But once you have introduced them, they become definite. So the first reference is "a tree T" but after that it becomes "the tree T". Now that you have named it, there is only one. (This assumes that you do not have two trees and call them both T. As in a context like this you're inventing the names, I assume you would not do this.) The same would apply in more conventional contexts. "I entered a room full of strangers. A [indefinite] tall man approached me. The [definite] tall man said ..." Once the context narrows the focus to an individual example, it becomes "the".
The vertex is "a" leaf because at this point in the context, the tree could have many leaves. So it is one of many. You could say that we are focused on one leaf in the sense that we have identified the leaf that is vertex v, but from a grammatical point of view, it is the vertex that is definite. The concept of leaf has just been introduced, and that is not definite yet. If a tree could only have one leaf, then it would be "the". Like: "As v has no parent, it is the [definite] root of the tree." As a tree can only have one root, we use the definite article.
I've probably just taken a simple subject and made it sound very complex. :-(
Best Answer
Copy is jargon in the editing/writing industry for the written word (text) - consider the word copywriter.
Web copy is therefore verbiage written specifically for use on the web.