Locust Tress
I have the same scenario along the edges of some of my fields.
I did the same for a few years. Now I treat it differently when I cut down a Locust along the edge of the field.
As soon as you cut the locust, or within about 45 minuets; spray a 50/50 mix of Roundup/water on the stump. Make the stump wet, especially on the edges.
If you still have some trouble with maybe a stump you missed and have root shoots from an older cut, use this chemical and procedure. Cut each shoot close to the ground, and put a few drops of TORDON on that cut. Though, make the cut wet and make sure the edges are wet, but a few drops goes a long way. It will have a blue color, so you will see this.
The TORDON is about $15 a quart, but it really goes a long way.
It is better to treat the cut at the time of the first cutting. Believe me, I learned my lesson.