From Jeffrey Scott Beckerman, Esq. of BECKERMAN & BECKERMAN in South Orange:

NJ case law is clear on whether you, as tree owner, have liability for damage caused to a neighbor's person or property.  The answer is a resounding "no" unless the neighbor can prove that you knew or should have known of the likelihood that the tree would fall.  For example, if you have a visibly rotting tree or tree leaning at a precarious angle, you may be considered to have an affirmative obligation to take the tree down and failure to do so may create liability for all damages.  In the absence of notice, the neighbor has no legal right to seek monetary compensation from you. Whether you choose to pay some money toward their homeowner's deductible is an ethical issue left up to you.

However, removal and cleanup of your tree on the neighbor's property is solely your obligation and failure to remove debris in a timely manner may result in a claim for damages if the continuing presence of the tree causes additional damages.  That claim would be only for damages caused by the delay, not for the initial damage caused by the falling tree.

If anyone thinks that a claim against Millburn Township might be worth pursuing for not inspecting shade trees for structural integrity, Polito vs. Millburn Twp. et al., Docket A-5923-09T4 [App. Div. 2011] will put an end to that idea. In order for Millburn to be liable, you need to show that the homeownerplaced Millburn or the Millburn Township Advisory Shade Tree Commission on notice of an anticipated problem with a specific tree, allowing an agent of Millburn [usually the Township forester] adequate time for the inspection to take place.  In the absence of those facts, Millburn has no liability.