Many plants have buds on their roots that will develop into shoots when the main tree is removed. There are several woody-brush killers-stump removers containing a chemical called triclopyr on the market that can be used to kill the suckers (and eventually the root system). A local nursery or garden center should carry one of these products.
Apply the product to the freshly cut stump of the nuisance trees. Only treat the stump and do not spread this product around. It will take a while to control all the suckers and kill the root system. Whenever you use pesticides, be sure to read and follow the directions on the container.