There's a good description at the PCA Alien Plants Working Group site. They say:
Common buckthorn is a shrub or small tree that can grow to 22 feet in height and have a trunk up to 10 inches wide. The crown shape of mature plants is spreading and irregular. The bark is gray to brown, rough textured when mature and may be confused with that of plum trees in the genus Prunus. When cut, the inner bark is yellow and the heartwood, pink to orange.
but they also note:
Several native American buckthorns that occur in the eastern U.S. could be confused with the exotic species. If in doubt, consult with a knowledgeable botanist to get an accurate identification. Carolina buckthorn (Rhamnus caroliniana), is a lovely native shrub that has finely toothed leaves somewhat resembling those of black cherry, and are smooth on the underside; it produces attractive fruits from August to October. Alder buckthorn (Rhamnus alnifolia), is a low-growing shrub that may grow to a maximum of 3 feet in height, and has leaves with 6-7 pairs of veins.

Here is a link that might be useful: PCA Alien Plant WG - buckthorn

I have invasive glossy buckthorn, acres and acres of it. I pull the small ones (up to about my height in loose soil) with a pair of large pliers - easier on my hands. The larger ones I cut down and stump treat with Roundup or do several rounds of spray or with "the glove of death*" each time they resprout. (* two layers of medical or latex gloves with a cotton glove with Roundup on it and wipe it onto the leaves with the glove.)

They are awful where there has been any disturbance - mowing, logging, etc. I'm considering it a lifetime task to at least keep them from increasing!

I've been battling the 2 invasive species of Buckthorn for several years -

- Frangula alnus, Glossy or European Buckthorn, aka Rhamnus frangula
- Rhamnus cathartica, Common Buckthorn

Both grow in a multi-stemmed shrubby habit, but Common Buckthorn has a tendency to grow in a more tree-like form. Common Buckthorn is dioecious, so only the female plants make berries. Seedlings are relatively easy to pull, but don't even bother trying to dig out med-large plants - the roots are embedded like concrete. Even the weed wrench is challenging with anything over 1 inch diameter.

I cut the stems close to the base and use a squeeze bottle with a concentrated herbicide (Glyphosate, Triclpyr, or 2-4-d). Apply herbicide immediately after cutting. If it is a large stem (2-4") is only necessary to apply herbicide in the outer edge of the cut trunk.

The plants do not have to be cut down, they can be left in place to die using other methods of applying herbicide. Cut bark treatment - where the bark and cambium are girdled or cut downward in a "cup" pattern all around the stem and herbicide is applied in the cup. Also basal bark application involves spraying oil-based herbicide over the lower couple feet of stems.

I've tried to eliminate the larger plants first, to prevent additional generations of seeds from being dispersed by birds. Don't have a good pic of Common Buckthorn, but here's one of Frangula alnus, Glossy Buckthorn -

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