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