yep buckthorn, dig it up and burn it, a nasty invasive weed species. Burn them all. Or turn them into bonsai. Since I kill a lot of trees I'm trying to turn into bonsai the end result will be as good as burning them. Nasty weed tree.
Probably okay as bonsai - Sorce has one, I've seen a few others. I think the timing one uses for Euonymous alata for pruning etc will work for Buckthorn. Older trees will have nice black bark. Leaves reduce well, berries are a nice fall accent, but flowers are insignificant.
My knowledge on this is second hand, internet knowledge and seeing "Beer City Snake's" buckthorn, I personally refuse to put effort into keeping one alive. I've seen too much damage to old prairie patches and woodland habitat caused by buckthorn, can't stand the thought of deliberately trying to keep one alive.