Imagine you were an ent, a tree person, and the day is going just fine, until.... TERMITES!!
You panic, stumbling through the glen as you try to scratch the annoying, horrifying and perhaps even deadly infestation out of your hide. Sitting in the stream to waterlog yourself, hoping they drown, you imagine a world WITHOUT termites.
What sorts of animals would a tree person decide are just much too alive for their liking? We can presume the ents has some limited magic, allowing them to manipulate the local food chain, perhaps having giant venus flytraps eat pesky animals. And they decide they're going to attempt to wipe out the ones that are "bad."
Termites and other insects which eat trees seem an obvious one which a forest would want to get rid of, don't they? Then again, as Dutch joked, that's going to leave a lot of dead wood. How about pigs, which tend to cause ecological damage?
The trick is, when you take something out of the ecosystem, even if it seems bad... there can be consequences. The ents might find that many of their plans to improve the forest nearly destroy it, similar to human efforts at tweaking ecosystems.
For example, if they brought in a ton of insect eating birds to get rid of annoying bugs, they'd also eat a lot of the pollinating insects like bees.
So, if you were a tree person, and had the power to wipe out various species of animals... which ones would you remove? Termites, pigs, deer, rats?