Ask Mr. Smarty Plants

Moraine locust is apparently a patented selected cultivar of Gleditsia triacanthos (honeylocust). The cultivar lacks the thorns that are so vicious on the original plant, have smaller seed pods and therefore less mess on the ground. They are susceptible to several pests and diseases and can become a weed problem and invasive. Obviously, you already know that since you have had it cut down. Now to get rid of the suckers that are still coming up. They are the plant's last-ditch struggle to survive, which is every organism's primary purpose.  Those "little trees" coming off the roots are, in effect, branches the roots are putting out to get some leaves growing to manufacture food for the roots through photosynthesis. The roots are a storage area for nutrients and will continue to try to stay alive. Of course, you want to nip off those suckiers as soon as they stick their heads up. Eventually, if you got them all, the roots would starve, but this could take a while. 

So, take the battle to the root. Get a bottle of wide-spectrum herbicide and some disposable foam paintbrushes. Dig down to where you can get at the main root. Cut off a slice of that root, exposing a fresh surface. Quickly, within five minutes, before the tree can begin to heal itself, paint the raw surface with the herbicide. This should go down into the circulatory system of the root and begin to kill it. This will also take a while, depending on how much root there is. If you are really determined, you could dig down and locate some more roots going out from the main stump and treat them the same way. Be very careful with the herbicide, avoid spilling it on the ground and contaminating the soil, and don't use spray. A spray could easily stray onto another, more valuable plant, and cause more damage than you intended. Hopefully, this will stop the development of the suckers and permit you to begin planting in the fall. We would also suggest that you remove as much of the root from the ground as possible, and then work some compost into the bed before starting to put in new plants.