Tree of Heaven
This is one of the toughest trees on the planet. Cutting it down actually stimulates it to resprout from the trunk and roots. (Some other methods of killing it reduce the amount of resuckering problems.)
The best method to kill suckers is to spray them as soon as you spot them. Use a herbicide containing triclopyr. (Glyphosate, the ingredient in Round-Up type products, is NOT effective.)
Read over this terrific fact sheet from Weeds Gone Wild, a website that covers all the ways to kill these obnoxious invasive plants: https://www.nps.gov/plants/alien/fact/aial1.htm
Getting rid of Tree of Heaven is a battle. It will take time and require vigilance to defeat this plant. The most important thing is to simply not let the suckers' leaves carry on photosynthesis and rebuild the root's reserves. You want to exhaust the roots. Each time the plant puts out new leaves and suckers, the tree's remaining roots have to use up part of their root reserves (stored energy in the form of carbohydrates.) So, if you kill the new suckers, you will eventually use up all the root's energy and all parts of it will die.
Next time you need to remove a tree of heaven, you may want to spray the entire tree so that the herbicide goes down and kills the entire (or at least almost all of) the root system. It's less work ultimately. And as long as the herbicide does not get on any other plants, the amount of herbicide use is limited. To use this method, also read the Weeds Gone Wild fact sheet. It discusses how to do each killing method.
There is more information about the issue of non-native Invasive Plants on our website.