Good evening and thank you for reaching out to ask2.extension.

I googled natural buckthorn control and found an article from the MN State Horticultural Society.  the following is from that article.

Once the buckthorn is uprooted, this is the best time to replant with native species. While the soil is disturbed, plant and seed to reclaim the space for native wildlife. Buckthorn are an understory tree that provide berries for birds. Replacing them with plants their equal in size and rival in ecosystem services is the long-term method of keeping the buckies from retaking the woods. Good replacements for buckthorn include dogwood, viburnum, snowberry, serviceberry, mountain ash, redbud, mulberry, sumac, leatherwood, and hazelnut. Always install new understory trees with ground cover perennials such as spreading native grasses, ferns, sedges, and native woodland flowers. Pennsylvania sedge, fescue grass, columbine, woodland phlox, wood sedge, and Solomon’s seal are a few native favorites sure to please the pollinators and help keep baby buckthorns from taking over again.   

I hope this helps.

I am not real sure. The plants above are from an article on what survives.  That doesn’t mean other things would not.  It will be a trial and error.  Your soil will be a factor as well.  So much of gardening is an unknown.  Sorry I can’t be more exact.  I don’t deal with buckthorn where I live, so I don’t have an personal experience.