There is a buckthorn thread in the native plants forum. It's been going for years. If anyone has a good answer I'm sure they have it over there. I think once they are cut, you can paint the stub with a certain herbiside to kill them. I try to pull them when they are small, and the root still pulls out easily. But I have a few that were too big to pull, that I cut comepletely down. I still need to paint them, but it must be applied to a fresh cut, so I will have to recut before I paint them.