The buckthorn is easy to spot right now with it's green leaves hanging on longer that pretty much anything else. And chances are, it's in your backyard. it's an invasive plant that grows so rampant that all the native and beneficial plants are choked out.
Renting a buckthorn puller from your city is a great way to pull it out from the roots, using no herbicide.
Girdling the big ones that are too big to pull is also chemical-free.
As a last resort, trees can be cut low to the ground and carefully painted with a stump-killing herbicide. Mixing in bluing from the laundry aisle can help you know which trees have been treated and where to avoid tripping on stumps.
If the job seems overwhelming, start by removing the female trees. They are the ones with little black berries on them. That prevents the spread of new seedlings.