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While New Jersey courts have upheld a landowner's right to use self help to trim encroaching limbs to the property line, the courts have not addressed self help in a root situation. While self help may or may not be available, the courts have upheld damages against property owners who permit their trees to encroach and damage a neighboring property. There may be a different standard concerning roots since the tree owner in many situations may not have any notice that the roots are encroaching the neighbor property, but in your situation the neighbor has actual notice of the roots.
So what does this mean?
Your best bet is to take good photographs that depict the problem and take them to a local attorney. For a modest fee you should have the attorney send a certified letter to your neighbor demanding the removal of the roots. Hopefully this will bring them to the negotiation table without the necessity of filing a lawsuit. Based on the photos you take and the response of your neighbor, your attorney will then advise the next step.
I would not exercise self help until a local attorney has reviewed the property and given you advise. The danger, even if the courts decide self help is permissible with root encroachment, is that the tree could die as a result of the root damage. It is generally recognized that you can trim encroaching limbs, but if you damage the tree itself then you could be liable for the damage. In other words your neighbor could sue if you kill the tree.
Please reply if I can help further.
Any general practitioner would be fine. I recommend you ask family, friends, and coworkers for a referral. The best way to find a good lawyer is by referral. If that is not available you can check an online directory such as http://www.lawyers.com/ where you can search by practice area (civil litigation or real estate litigation) and geographical location.