The general rule in Pennsylvania is that landowners are NOT responsible for limbs that overhang their neighbors' property. In other words, you cannot compel him to pay a tree surgeon to come out and trim the branches on your side of the property line.
On the other hand, you do not need permission to cut off the branches that overhang your own property; if you don't like the shade they cast, or if you worry about safety, you can hire a tree surgeon yourself to come and cut the branches off.
Cutting overhanging limbs, however, presents a Merchant of Venice problem: you can cut the limb at the property line, but YOU ARE NOT PERMITTED TO TRESPASS ON HIS PROPERTY. That means that you cannot cut the limb on his side of the property line, and cannot set up a ladder there to do the cutting.
I STRONGLY suggest that, before you decide to cut down the limbs, you sit down with your neighbor and discuss the matter. Although you have a legal right to cut the limb at the property's edge, a discussion with him might reveal that the property line is not where you thought. Alternatively, he might not want you to cut the branch off and, in compromise, could offer to pay for maintenance himself in exchange for less drastic trimming. Regardless, giving your neighbor a heads up before you mutilate his tree is good for your long-term relationship with him.
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