It would help to know the original poster's location. Sounds like Tree of Heaven (Ailanthus altissima), it is known for vigorous root sprouting after being cut.

White Poplar is another possibility.

Resin

Most trees seem to do the same thing, and some even do it worse like you describe yours as doing.

The problem with wacking off the root suckers is the fact that the healthy root is then motivated to flush many more suckers up in a its bid for survival.

Most trees which are first cut down, can have holes drilled into the trunk and a lot of the following mixture, I am about to describe, saturated all over the trunk. Then Schedule some a professional to a couple days later and gring the trunk and all the major roots growing out of that trunk. I used that method on some very tenacious hackberry trees I had cut down, and the sucker type problem never occured.

Since at the time your trees were first cut down, you did not know to do the above; the only way I know how to deal with such problems is to get an effective, shrub and brush killer, pour it into a non corrosive bucket along with an equal amount of diesel fuel.

Then use a Paint type brush to apply the mixture to the sucker stems and leaves growing out of the root. Also if any of the orignal tree's roots or trunk can be seen above ground, drill holes in them and immediately apply the stuff to them with a brush that has been saturated with the mixture.

The use of a brush to apply the mixture is the easiest way to control not getting the mixture on the ground or oversprayed to other areas you do not want contaminated or killed.

Apply the mixture in a systematic way, so that you do not have to rub up against areas you have already treated to reach other areas which still need treating.

Be careful not to get any of the mixture on your skin, and wear covering over your clothing which can be washed immediately, or thrown away. Do not let any of the mixture splash on your face or skin. If that does accidently happen wash the affected area and pat dry immediatly.

Take a complete and effective Shower when You finish the task. But just prior to entering the shower put any clothing you wore that you did not want to throw away into the washer with no other cloths added. They can be washing while you shower. Once the washer finishes lay or hang the just washed clothing outside to dry. Do not put the cloths into the dryer, just in case some of the fuel fumes did not get completely washed out of the clothing.

I know that all sounds like a lot of trouble, but it is worth the effort just to stop the invasion.

By all means do not give in to the temptation to put the mixture into a sprayer, because that would contaminate the soil, along with any watering or rain runoff. Not to mention the contamination, which would occur from overcast spray hitting your other prized plants, shrubs, and trees.