First, you should consult with your neighbor by verbally asking your neighbor in a nice way to trim the tree. If the neighbor refuses or neglects to trim the tree, then you should send the neighbor a certified return receipt letter again requesting that the tree be trimmed as the tree is causing a nuisance (leaves falling onto your roof and yard) or that the tree poses an acute and chronic danger (damage to property and injury to person). It would also be helpful to obtain a letter from an arborist or tree expert confirming that the tree needs to be trimmed or removed by your neighbor. A copy of the tree expert's letter could then be included with your letter to the neighbor warning the neighbor that you intend to cut the tree branches if no action is taken and that you may seek reimbursement for the cost of cutting or maintaining the neighbor's tree. If the neighbor still refuses to take any action to perform maintainance related to the tree, then you may cut the tree branches yourself that extend onto your property. If you are required to hire a tree trimming company you may also sue the neighbor in small claims court for reimbursement depending on the jurisdictional limit in your state. By way of example, California allows an individual to sue for amounts up to $10,000.00 in small claims court. Remember to take lots of photographs to document the problem with the tree overhanging your roof. Of course, you might get sued by your neighbor in small claims court for cutting the tree branches. The key is to create a paper trail of letters requesting the neighbor to maintain the tree and have photos of the tree so you have evidence to show the court, if necessary, that you were trying to be a resasonable and good neighbor. Judges apply the "reasonable person" standard with respect to negligence claims especially when a dispute errupts between neighbors regarding a poorly maintained tree.
For example, if a wind storm were to blow the tree down on your roof, then you would have proof of the neighbor's negligence and could sue the neighbor for your damages. I have seen cases where a homeowner fails to maintain a large tree on his property and that tree later falls over in a wind storm and causes tremendous damage to his neighbor's home and cars. Proof that the tree was sick or poorly maintained becomes critical in order to obtain damages. Otherwise, the incident could be deemed a force majeure or "Act of God" (non-negligent act on the part of the landowner with the tree) and the damage to your roof will be your responsibility and hopefully covered by your homeowners insurance policy (less your deductible). Good luck!
The short answer is likely yes. But disputes between neighbors can start from the smallest event or lack of communication, and can grow into uncomfortable, lengthy, and expensive feuds. So, I strongly suggest that before cutting anything, you first talk with your neighbor to address your concerns and intent regarding the overhanging branch.