Lori, sadly some trees and plants are considered invasive and all you can do is try and keep ahead of it! Some are Mimosa, redbud, althea (rose of sharron), nandina, etc. You can either spray them or pull them up early on before they develop a root system. A neighbor had a mimosa in his yard and we rejoiced when it died some 25 years ago! We still get seedlings in our yard from somewhere! We scream at them and pull them up. If we find them large, we spray them! Wish I had a better answer but as long as you have an invasive tree, you will have this problem. You may want to get rid of the main tree and plant something else. Sorry!
Looks like grumbling and yanking, because this tree is just too beautiful to get rid of.
Do you have any other plants in the area that you would like to continue to reseed? If not you can try a pre emergent in the area that may help. I will do a little research and see if there is one that will prevent the seeds from germinating. Otherwise it will be using your fingers to yank them out.
Four Seasons--I am okay yanking in flower beds, but the lawn bugs the dickens out of me. Would a pre-emergent ruin the grass?