Trees and neighbors...advice so you don't get sued
This article provides some great insight into the laws regarding tree ownership. Remember to check your neighborhood laws before following these!
One of my biggest take aways from this article is if your neighbors tree has branches hanging in your yard, you are allowed to prune them back to your property line.
Another interesting thing I learned was about "boundary trees". This is when the base of the tree sits directly on the property line therefore both property owners share in the responsibility of the tree.
A link to the full article is below, there's a ton of great information about tree responsibility.
Source: Brandon Gallagher Watson, Trees and Neighbors and the Law, Arbor Age.