The police have no interest in civil matters. This is a civil matter.
You problem in proceeding is that technically, the common law says that if a tree on one property is overhanging another, the property over which it overhangs is allowed to cut the branch (at the property line) so long as it doesn't damage the tree and lead to the death of the tree. So even though the branch only crossed the neighbor's property for the size of a
five gallon bucket, , it did cross their property line according to them.
If you can't just let it go, your first action is to have a survey done to determine if that
five gallon bucket, of the branch was indeed over the land of the neighbor. And if it was, then you could still file a lawsuit based on the impact on the neighbor's land and yours and see what the courts have to say assuming you have deep pockets to try to establish some new standards.
If the branch did not cross entirely onto the neighbor's property then you have s suit for trespass. And if the branch was both on your property and the neighbor's property at the point it was cut, then you were joint tenants and the neighbor would need your permission to cut the branch. Without your permission, they could be held to a crime.
http://codes.ohio.gov/orc/901.51
If you have the resources to proceed or the will of principle to make it right, then you have to choose the next step. Or you walk away.