Generally speaking, if there is any possibility that an overhanging limb could cause a safety issue or damage to property (i.e. it might fall on his house/car/jacuzzi in a windstorm), you have an obligation to cut it to prevent that. Additionally, he can cut your tree to the extent it hangs over his property, but can't enter your property to do so without your permission, and can't damage your tree (i.e. if he kills it you'd be able to sue him for whatever the "value" of the tree was). So no, you don't have an obligation per se to prune the tree.
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