A good dose of broadleaf weed killer containing triclopyr will finish off most trees. Apply it to the cambium of the stump after cutting the tree down. It won't even make your lawn nervous, either, so you can also treat the sprouts coming up anywhere in your yard. This is good stuff to treat those weed trees coming up in the fence in the back yard, too.
Do it right after you cut the tree down. Don't wait even a half hour. The tree shuts down transferring nutrients to the roots pretty quick.
I find that a concentration of Triclopyr stronger than the usual 1-1.5% used for spraying foliage works a lot better for this. Your garden store stuff will likely be 1-1.5%. That works great on leaves though, when the plants are growing. Triclopyr is a growth disrupter so it needs the plants to be growing.