If my neighbor's tree branches hang over my yard, can I trim them?

In a New York Times article entitled ‘Talking Trees”, the author states that “lawyers agree that certain basic legal tenets will apply in any state. For example, the ownership is normally determined by the trunk. If it is clearly inside a property line then the tree belongs to the owner of that property regardless of how heavily its limbs hang over the neighbor's land.”

Let’s assume your neighbor has exercised his/her rights to trim your tree up to your property line. What should you do if your neighbor demands reimbursement for the costs she incurred to trim back the branches? According to attorney Marc Campanella, “Unless the tree is posing a hazard to your neighbor’s person or property (yes, that standard is somewhat open to interpretation), the law does not require you to personally trim the branches or reimburse your neighbor for any expenses he/she might have incurred from electing to trim back the tree herself.” 

My neighbor damaged a tree on my property.

Nolo legal reference advises that if your neighbor or someone else cuts down, removes, or hurts a tree on your property without your permission, that person is required to compensate you (the tree owner) for your loss.