In general, a property owner is responsible for trimming trees on that occupy space on their property. Thus, you were not responsible for trimming portions of your tree hanging over your neighbors property line. The exception to this rule is if you have notice that your tree has become a hazard (ie. large limbs fell during a storm and appear more will fall if you don't do something).
Likewise, your neighbor has the right to trim your tree up to their property line. Technically, the can not enter your property to assist in this maneuver and may not trim any branch nor portion that does not over hang their property. Thus, they trespassed when they entered your land and caused damage by removing the tree.
An action could be brought for private nuisance by trespass, as adopted in Oklahoma from the Second Restatement of Torts:
Liability for Intentional Intrusions on Land, provides:
quote:
One is subject to liability to another for trespass, irrespective of whether he thereby causes harm to any legally protected interest of the other, if he intentionally
(a) enters land in the possession of the other, or causes a thing or a third person to do so, or
(b) remains on the land, or
(c) fails to remove from the land a thing which he is under a duty to remove.
Angier v. Mathews Exploration Corp., 1995 OK CIV APP 109, 905 P.2d 826
HOWEVER... the damages would be the financial loss you sustain as a result. BURKE v. THOMAS
1957 OK 154. If the tree were killed, you could argue for reduced value to your property OR the replacement cost of the tree. Since it is not killed, all you can argue is the reduced value of your home. Which is probably negligible (damages are on a property basis or agricultural basis on trees, either way you're screwed).
I did a case where a fly-over spray to kill trees in a pasture destroyed the trees on a 60 acre rural homestead. All the law I was able to find set the damages on what the economic or sale value of the property was before and after. A rural tract of land in no-where Oklahoma is worth more as cleared pasture than it is as scrub forest even though the owner wanted the scrub for the few fruit trees & pecans as well as recreational and aesthetic uses. We settled for the cost of clearing the dead trees calling them safety hazards.
What I'm getting at is they did you wrong... but you have no real recourse. While it was intentionally (most cases are accidental) there are no real damages to take action on. Perhaps you could demand that they hire a forestry expert to take proper care of the tree or at least ask them to finish the job and apologize.
I'll see if there is a city ordinance or anything later...
/end crappy and quick legal research mostly from memory