There are two ways to pronounce that most common of words in the English language, the. Despite what you might have heard, the does not always rhyme with me and tree, and its pronunciation has nothing to do with verbs and nouns.
Here are the facts: the standard way the word is pronounced has to do with what follows it. If the word that follows the begins with a consonant sound, then it's pronounced \t͟hə\—that is, with the same vowel sound as the word of—and it is unstressed:
the one (Note that although "o" is a vowel, the word one begins with the sound usually made by "w," a consonant.)
If the word that follows it begins with a vowel sound, the is usually pronounced \t͟hē\ (to rhyme with me), but is also sometimes pronounced \t͟hə\, as above. Again, it is unstressed:
Occasionally the is stressed, for instance to provide emphasis, or before a long pause. In those cases it's pronounced \t͟hē\:
the one and only
the queen of hip-hop
If you're a native speaker of English, this information only confirms what you've been hearing and likely doing all your life. And as linguist Mark Liberman points out in a post at Language Log (a group blog about all things linguistic), what we hear today is what our grandparents heard all their lives, and their grandparents before them, in both American and British English.