You don't have much to worry about. That doesn't mean your neighbor doesn't sue you (because anyone can sue anyone if they want)... but your neighbor doesn't have much of a leg to stand on. But while we are talking about trees, I thought I'd elaborate a bit here for others.
First off, it is legal for a homeowner to cut branches that trespass from a neighbor's property, provided that the homeowner uses reasonable care in their removal and does not become a trespasser in the process.
Second, If damage has been caused by trespassing trees or the removal of the offending branches will be costly, a homeowner/neighbor can seek recourse by suing in small claims court for damages done to his or her property. An example here would be if you planted a tree on the property line and then the roots from the tree damage your neighbor's home's foundation.
In your situation, I do not think there were any damages. When the branch fell, it didn't damage the land. It might be a different answer had it damaged your neighbor's house. Then we would need to do a analysis if you were negligent and owed some duty of care to your neighbor. That's a topic for another day.
Typically a person (ie your neighbor) is responsible to remove the debris from his yard if it happened due to a natural occurrence. For example, let's just assume your property backs to one of the regions forest preserves. If a branch falls from a tree from the forest preserve on to your land, you aren't going to sue (and win) any kind of judgment from the State or Park District for the cost of removing the branch.
Tell your neighbor that they may be covered under his own homeowner's policy for the damages and removal, but that you aren't financially responsible for naturally occurring incidents such as this.
I try to provide an opinion, or what I would do in your situation, however, this answer is intended as informational only - based upon very limited facts - and does not constitute legal advice or form an attorney-client relationship between us. I do not claim to know every single law because laws are often changing. Therefore, I encourage you to consult with another attorney regarding your case and time limitations.
Your neighbor handing you a bill does not make your liable. Trust your insurance carrier. Nothing you did caused the damage to your neighbor's yard. Your neighbor is also free to sue you is she wants to, but the cost of the suit would be prohibitive and she should probably just let it go. As a gesture of goodwill, if you choose, you could offer to split the costs of her cleanup but you are not required to do so.
Just out curiosity, in the fall, does she rake the leaves, sort through them, and then send you a bill for your leaves? And I assume she has no trees that would dare litter her neighbors' yards with leaves and branches. The answers given were spot on--you are most certainly not liable. Unless you were aware that there was a dangerous condition caused by your trees and property damage or injury resulted, you have nothing to worry about, except having an annoying neighbor.
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