Well, it depends where you're at. My experience is that the first time the tree hugger just shows up, says it doesn't like your face (and tree chopping), and just leaves, in a rather abrupt (and confusing) fashion. The next time the bugger shows up, it'll demand you curb tree cutting. If you accept then, you have a deal (negotiated somewhat). If you refuse, I assume the'll come back armed and unreasonable (I've made deals with them so far), or possibly you can negotiate further: I haven't explored that branch. They're just taller goblins, though, but with crap wooden spoils...
If you made a deal, it says the allotment is until it comes back, which should be in a year (spring next year). I chop down most of the quota immediately, and then leave a dozen or so trees for when I need to get rid of a tree that's in the way, finishing off the quota as spring arrives. So far I haven't exceeded the quota, so I don't know what happens then.
I start out chopping down anything at sight to build up a buffer before the tree huggers start making demands. Running any kind of metal industry with the tree hugger tree allotment is going to require a very careful handling of where you use your wood (if you don't have access to coal in some form). Magma will allow you to handle that, but it takes some time to get there in a controlled manner unless you're in an area with easy magma access.
However, Skuggen's advice is sound: build defenses. Also, a construction where you can retreat into your fortress and combine waiting the enemy out and building new killing devices for them is a good thing (My first attacker was a fire breathing titan. While I hid from that one, working to kill it, a forgotten beast entered the cavern. I sealed myself off from that direction as well, and eventually the forgotten beast got killed from fighting all the other wildlife entering the cavern, but the titan I had to deal with myself). If you've got goblin neighbors you don't have to kill tree huggers to get some action...
Well, Astrid's bloodthirstiness is to be admired, and a long sealable corridor with suitable traps is the normal starting point. Personally I'm fond of lever cranked menacing spike traps used on the trapped enemies.