Articles are generally only needed if they refer to a single object or quantity. "Drink the milk." is a valid sentence as you noted, indicating that there is some obvious milk available. "Drink a glass of milk." Is also valid, specifying an amount but if there are multiple sources of milk available any of them is fine.
Removing the article does not change the structure of the sentence, and it remains equally valid. "Drink milk." with no article doesn't provide any restrictions on which milk is drunk or how much milk is drunk, but it certainly is a valid imperative commanding the recipient that they should drink milk; technically any milk, even a small sip, would fulfill the command.
A slight complication comes from "countable" versus "uncountable" nouns, as with your "Cut trees." example. For countable nouns, the plural form is used ("trees") and there is an extra implication that it should be more than one. I would categorize the options as follows:
- "Cut a tree": Any tree is fine, but cut just one.
- "Cut the tree": There is a single valid tree, and it must be cut.
- "Cut trees": Any trees are fine, but cut more than one.
- "Cut the trees": There are multiple valid trees, and they all must be cut.
The uncountable version has only minor differences
- "Drink a(n) [amount of] milk": an amount must be specified, since "one milk" isn't clear, and any source of that milk is permitted.
- "Drink the milk": There is some specific quantity of milk clear from context, it must be drunk.
- "Drink milk": Any source and quantity is permitted, though there is still some indication that it should be significant.
- "Drink the milk": Uncountable nouns don't have a separate "specific items, large quantity" format, so this is identical to "specific items, small quantity"