The branches need to be taken back at least to a side branch 1/3 or greater the diameter of the main branch being trimmed. This allows the tree to cover over the damaged area in a reasonable time and hopefully prevent rot. Never leave a stub sticking out. The tree won't be able to close off the wound quickly and rot will occur. Removal of the entire limb, back to the trunk, is often best for the health of the tree.

Never trim limbs back halfway. Trim them to the branch bark collar. If you do not know what this is, please read something on proper tree pruning before you begin, and ensure you make your three cuts on each limb, the first being the undercut.

Dan

generally speaking.. you own the air over fence..

you generally have a right to trim back to the property line ...

a smart neighbor.. would allow you to come in to his yard ... to trim them back to the trunk .. which is presumably not on the property line...

you are on your own to figure out what NY state law is ...

good luck

ken

Dan brings up a couple of good points. Be sure to understand how to make the cuts. It's usually best to remove the bulk of the limb first and then come back and make the final cut. Making the undercut first prevents the limb from tearing back into the main branch or trunk. If the limb rips a chunk out of the branch collar, it can take much longer for the tree to cover over the wound and rot is much more likely to be introduced.

If you aren't familiar with the terms "branch bark ridge" and "branch collar", take time to look them up and make your final pruning cut immediately beyond them without cutting into them. The tree's ability to cover over the wounds quickly is maximized with proper cuts.