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In general, the location of the tree trunk determines who owns the tree. A tree that stands solely on your client's property belongs to your client. Disputes arise when trees straddle a boundary line or when the branches of your client's tree encroach onto the neighbor's property. Jurisdictions differ on boundary trees. In some states, trees standing along a boundary line are the common property of the neighbors on either side of the boundary, and neither neighbor can remove the tree without the consent of the other. This includes the tree that starts out in one yard and grows into the boundary of the neighbor's yard.
The problem in Ohio is there are few clear cut laws on this. The law says you can cut the branches back as long as it is not endangering the life of the tree...hard to prove and often these cases wind up in front of the judge.
To prevent being held liable in court you may want to talk to the owner. If they refuse to do anything then consider filing a petition with the courts asking that the owners either cut it back or allow you to do so. By going though the courts first you avoid being charged with the replacement of the tree should you neighbor allege th tree was damaged.