Generally, trees grow randomly next to each other. So if you chop down the very last tree, you messed up.
They have a maximum distance for new trees and a maximum amount of trees in that distance. They will reproduce faster when less trees are around. (So a single tree can grow into a small forest quickly, but a large forest won't get too crowded).

They do not need water but they need to grow on soil material (like earth, sand, ashes, ...; NOT rock). But a pixel might be enough.

There are some plants that grow differently (cotton, big mushrooms, sprout berries). But the trees should behave more or less the same.
There is a little todo, though: currently places can get too crowded with different species of trees. For example: it /should/ be 10 trees total in an area; right now it is 10 oaks, 10 palm trees, 10 coniferous trees, ..)

PS: ah, and different maps can tell the trees where they are allowed. So there might be maps where trees grow only in a special area.