Ask Mr. Smarty Plants

Ilex vomitoria (yaupon), while a sturdy Texas native, can also get to be a pain when it is big and takes over an area. There are two alternatives to consider: Dig out the roots very thoroughly or go the cut and paint route. The second alternative involves slicing the top of the main root off very close to the soil. Using a disposal paintbrush, paint the cut surface with a wide spectrum herbicide. Do this very quickly, within 5 minutes of cutting, as the root will immediately begin to heal over in order to protect itself. The herbicide should get into the root system and begin to kill it and discourage the production of suckers. However, the primary objective of any plant is to survive, and whatever vestiges of root are left alive will continue to try to sprout, forming what are basically new branches to bear leaves and, through the process of photosynthesis, produce nutrition for the tree. 

After employing either (or both) methods, continue to pull out the suckers as quickly as they appear. Eventually, one way or another, the roots will starve to death and die. Be very careful with the herbicide, don't spill it on the ground or try to spray it. You can easily contaminate the soil or harm another, more desirable plant.