Picture an imaginary wall from the ground to the sky, on your boundary line. Whatever is on your side of the boundary, is yours to maintain. So, the branches that overhang onto your property are yours to do as you please. You see, if your neighbor came onto your property to trim that overhanding branch, that would be trespassing. And, if he hurt himself doing so, that would be a possible claim on your homeowner's insurance policy.
It is neighborly to tell your neighbor that the branch is a bother and you plan to trim it. Then do so and remove the debris yourself because it is yours now.