You can report this to the Town and see if they do anything such as issue a violation. You can also sue but I don't know what the value of a tree may be. Perhaps you can convince the neighbor to plant a new tree.
I am a former federal and State prosecutor and now handle criminal defense and personal injury/civil rights cases. Feel free to check out my web site and contact me at (212) 577-9797 or via email at Eric@RothsteinLawNY.com. I was named to the Super Lawyers list as one of the top attorneys in New York for 2012. No more than 5 percent of the lawyers in the state are selected by Super Lawyers. The above answer is for informational purposes only and not meant as legal advice.
I agree with Mr. Rothstein. You could sue - but given the amount in dispute - you might be better offer requiring the neighbor to plant a new tree.
A neighbor has not right to enter your property or cut down trees on your property.
If this answer was helpful, please mark it as helpful or as a best answer. This answer is for general education purposes only. It neither creates an attorney-client relationship nor provides legal guidance or advice. The answer is based on the limited information provided and the answer might be different had additional information been provided. You should consult an attorney.
The only think I could add is that there is a NYS law that would triple the "damages" related to removing the tree. Environmental Conservation Law.
You should seek specific advice about specific problems because legal analysis takes time and may change if the facts change.