Pruning a Buckthorn
How do I prune a four year old Buckthorn?
Chautauqua County New York
How and when you prune a plant depends on what you want to do with it. Here in Michigan, buckthorn is considered an invasive and we generally are trying to kill it.
However a quick search of the internet revealed several articles and videos on the subject. I suggest you search for these and decide which fits your needs and goals.