Some trees will come back with a vengeance if you just grind them to below grade and cover and do not kill the roots, Locust comes to mind but they're are others. If that is the type of tree you have cutting as close the ground as possible and then drilling a bunch of 1" holes about 12" deep in the stump then pouring Epson Salt and cap with melted wax from a candle. Cover with black plastic and then dirt. This will kill all of the roots and keep the tree from spouting again.
Pine, poplar and most oaks will not spout back from the stump so just cut as close to the ground as you can and if you do not want to grind the stump just cover with dirt and wait until it rots.
Some poplars certainly WILL come back. I took out 7 hybrid poplars early last spring and ground the stumps. I fought the roots all summer all over the yard. I got some strong brush killer and about twice a week all summer, I'd mix some with water 50/50 and slap it on the shoots with a paint brush. It didn't take much of it. It would knock them dead but more just kept coming up. I never let any get above 6". We'll see if they come back this year. With no leaves all summer, hopefully the roots have starved out by now.
