We cut down a 30 foot volunteer poplar in our yard that the idiot previous owners had allowed to grow in a bad spot - cut it down, ground out the stump, and just kept digging up the shoots that popped up. It’s done for good now (she says, tempting fate). It was beside a lilac, and maybe that helped - lilacs are bad-ass. 
[QUOTE=KRM]
Roundup (or a generic equivalent) is a contact herbicide and will not harm the soil. I’ve never tried to kill a tree with it but it should work. Be sure that it’s a very calm day as overspray would not be pretty. You also don’t want to mix it too strong or you will only kill the top part of the plant and it won’t get down into the roots.
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I’ve done this with Alianthus trees (tree-of-heaven), they look like Sumacs. They send out shoots from the roots when you cut them down too. I’ve soaked the leaves with Roundup, you can also cut into the trunk and spray some Roundup in there too. Cut it down when all the leaves turn brown. Then check it every spring, just in case it rises from the dead.