Conifers or greens such as Pine, Spruce, Cedar and Juniper will not sucker, and die off on their own once cut to the ground. Many Deciduous or leaf trees will generate suckers or sprouts from the stump or surface roots and subterraneous roots after the tree is removed.
There are two ways to deal with this (besides extracting the whole root ball, which is usually impractical and costly unless you are excavating).
Firstly, one can apply a herbicide such as Killex or Roundup to holes or slots cut in the stump, then cover the stump with plastic to prevent the herbicide from being washed out. I’ve heard from a couple of clients that pounding various 1” lengths of copper pipe into the stump along with the herbicide helps with stubborn species like Poplar. The herbicide will be absorbed into the root system and kill it. One may have to apply this a couple of times, and by next season, the stump and roots should be dead. Then the stump and surface roots can be removed if desired. The main drawback to this system is that adjoining vegetation may be adversely affected – that is, neighboring trees, shrubs and grass may be harmed by the poison in the entwined roots.
For this reason, we suggest you consider the second alternative: After the tree is removed, remove the stump and surface roots as well, then wait to see if any suckers come up from the remaining roots in the ground. When and if they do, wear rubber gloves and apply herbicide to the gloves, wiping the herbicide onto the opened sucker leaves. One will have to keep at this for up to a year or two before the root system suffocates and dies. This method will not affect adjoining vegetation nearly as much.