In most areas of the US, you have the right to trim trees that overhang into your airspace. You can also trim roots as needed, for example to put in a retaining wall. All this needs to be done with as little harm as possible to the neighbors trees. For example, if you cut major roots to the neighbors trees for no good reason and the tree then dies, he could bring legal action and wood likely prevail.
Any trees actually growing exactly on the property line are owned by both parties and both are responsible for any decisions, eg cutting them down, etc.
Some particular areas may have local laws which apply, but most places don't.