The Buckthorn is indeed invasive. A main cause of this is that it leaves out before most other species begin to leaf. By doing so, it out-competes the others for the vital sunlight. In the northern woods, it presents a further problem: it provides habitat for the invasive European earthworm that is damaging so many northern woods. In areas where the Buckthorn has been cleared, the invasive worm population has dropped by 50%.
Here is a "Fact Sheet" from your northern neighbors:
http://www.ontarioinvasiveplants.ca/files/unwantedlettersBuckthornFINAL3.pdf
I certainly would NOT suggest eliminating the species from your property, but its growth/spread should be monitored to assure that it does not get out of control...that would certainly attract the DNR people who believe poison is the best medicine.
In fact, the tree is very important in natural veterinary medicine. It is the active ingredient in the famous "Buckthorn Syrup", possibly the best known herbal purgative.
Buckthorn Syrup recipe:
(Can be made from the berries, or the bark. However, the bark must be properly dried and aged - at least a year, and even then, it can cause violent vomiting, especially in younger animals. Therefore, I only give the recipe using the berries.)
Simmer 5 ripe berries in 1 cup (½ pint) water for about half an hour (do not boil).
Wait for it to cool, then add 1 T honey and ¼ t ground ginger.
This yields one cup syrup, which is a single dose for a young, small animal such as goat or sheep.
One single dose should purge the animal. Try to make the animal fast for the day.
(Larger animals - cattle/horses - get double this dose)
The leaves are very high in Nitrogen, and decompose very quickly.