Tree animation on the map.

First, the game will have whatever is defined in it by the editor.

I disagree; I have animation disabled in the editor. But despite this, animation works in the game :)
Animation in the editor is just a visual effect that is not saved in the map. Regardless of whether animation is enabled or disabled in the editor, it will always be present in the game.

Secondly, the editor has Object Animation and Terrain Animation.

The question was not about the editor, but about the game. And it wasn't about the type of animation you're thinking of.