Thank you for your question. Although your pictures are very helpful, I would suggest a little more investigating. One way to identify common buckthorn is to scrape the bark to see if there is a thin orange layer of tissue below the surface. Another way to identify it is to look for a thorn at the tip of the branch surrounded by two buds that resemble the shape of a deer hoof. If there is no orange layer or thorns you might have a black cherry tree/shrub. Below are two links that will help you identify the plant in question. If you conclude that it is buckthorn the best way to remove this large specimen is to cut it down and treat the outer edge of the cut (cambium layer) with 20% glyphosate. You can use a small foam brush or bingo dauber to apply the glyphosate so that you only touch the plant you want to kill and no other. This is the perfect time to use glyphosate since the plant will take it in down to the roots. It doesn't work in the spring when the sap that runs up will simply push the poison right out of the cut. In the spring you can pull smaller specimens out of the ground. Included is also a link to learn more about using glyphosate.
Reminder: A pesticide label is a legal document. Always follow the pesticide label directions attached to the pesticide container you are using. Pesticide labels may change frequently. Internet labels may not match the label on the container you are using. The site of use or plant to which the pesticide is to be applied must be listed on the label or the pesticide cannot be used. Remember, the label is the law.