Ask Mr. Smarty Plants

Seedlings coming up beneath the tree and adventitious root sprouts ("suckers") are two different things. The seedlings have instructions in their seeds about making roots and starting to get nutrition from the soil. The adventitious sprouts have roots, all right, but it's the main tree root, itself. They are the root's fight for life. It is trying to get some more "branches" out to grow some leaves, get some sun and start performing photosynthesis to keep the roots alive. If you remove that sucker, you have removed it from its root. If there is still trunk remaining above the ground, you could try the painting method. Get a disposable foam paintbrush and some full-strength broad spectrum herbicide. Cut across the remaining trunk and immediately (within 5 minutes) paint the cut surface with the herbicide. Be very careful doing this, don't allow it to get on the soil or any other plants. The herbicide needs to go on the the cut surface quickly because the roots will start to heal over to protect themselves. The suckers should be removed by digging down several inches and prying them out. Until the roots have been starved to death, you will still get the suckers. 

Here is some related information, Prune for the Love of Live Oaks from your own local newspaper, the Standard-Times, gosanangelo.com, e-edition.  It has some information on suckers.