Hello Edward,

Managing buckhorn is a perennial project. Persistence is key as plants from seeds will be a continual issue, dropped by birds and other wildlife. There are several methods used to manage buckthorn depending on location, stage of growth, and time of year. Below, I am giving you extension articles that describe management options. 

The baggie technique is used on cut stumps and is relatively new. Non-chemical control options for cut stumps include covering the cut stump with a tin can or durable black plastic (such as a "Buckthorn Baggie") to prevent re-sprouting. After cutting the tree, apply the can or plastic over the cut stump and root flare.  Use nails to affix the can or a tie to affix the black plastic. Leave in place for one to two years.

http://files.dnr.state.mn.us/natural_resources/invasives/terrestrialplants/woodyplants/buckthorn_what_you_should_know.pdf

https://news.wisc.edu/buckthorn-baggie-kills-invasive-trees-without-chemicals/

https://cpb-us-e1.wpmucdn.com/blogs.cornell.edu/dist/b/5769/files/2015/12/Buckthorn-1s2q4e9.pdf

https://mnfi.anr.msu.edu/invasive-species/GlossyBuckthornBCP.pdf

If you have more questions, please write again. Thank you and good luck with your project.