In most russian grammar books there is a rule saying that definite article must be used with superlative adjectives. However from time to time I see people using indefinite article. For example, a title of the film "A Most Wanted Man". Can somebody explain this?
Learn English – use indefinite article with superlative adjectives
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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. :-(
Your description of the use of the definite article as opposed to the indefinite article is accurate. The presence or absence of one of these articles, however, in no way alters that standard usage.
As for why the article may disappear, it strikes me that when the article is there, it draws attention towards the adjective and away from the person. Let me use sentences 4 and 5 to illustrate:
It seemed Walter didn't pay any attention to a tearful Kitty.
Stated this way, the sentence indicates that Walter is not paying attention to Kitty's tearfulness; he is not moved or affected by her distress.
She hasn't got lovely eyelashes like marvelous Monica.
In this case, the attention of the sentence is squarely on Monica, and "marvelous" is relegated to being a pure adjective, neither calling for nor receiving any unusual or special notice from the speaker. If we added "the," it would put the attention on the fact that Monica is marvelous. (This can, by the way, be a vehicle for sarcasm, if you want to point out that she's really not all that marvelous after all.)
Best Answer
Generally, no. As mentioned in the comments, phrases like "a tallest mountain" or "a biggest dog" do not sound sensible.
However, the use of "most" as an adverb does not necessarily imply the superlative; it can be merely intensive. From Merriam-Webster on most:
The first definition implies the superlative, but the second definition is merely an intensifier. The phrase "a most wanted man" (by the second definition) is equivalent in meaning to "a very wanted man".