Cutting down a tree is not always enough to get rid of it. Some invasive tree species send up suckers or shoots that will eventually grow into adult trees. The tree's roots are responsible for sending up clone trees all over the yard. One way to stop the tree's invasive growth is to poison its roots to kill the tree. Take care because the systemic herbicide you must use to poison the tree's roots can also weaken any tree whose roots grow near the target tree's.
Girdle the tree. Cut a groove that encircles the trunk. Make the groove 1/2-inch deep on small trees (1 to 2 inch trunk diameter) and up to 1 1/2 inches deep on large trees (6 inches in diameter or more). Make the girdle 1 to 2 inches wide on small trees or 6 to 8 inches wide on large trees. To make the cut with an ax or hatchet, make a notch above and below the girdling line to remove a plug. Repeat around the girdling line.
Spray or paint a systemic herbicide on the girdle in the tree as soon as possible, ideally after you make each notch in the girdle. Thoroughly saturate the girdle but do not apply enough to cause run-off.
Watch the tree's root zone carefully for suckers or sprouts. Spray them with the herbicide as soon as you spot them.
Cut the tree down once the foliage dies, and the tree produces no more suckers (this may take as long as a full growing season). Once you poison and kill the roots, they will decompose over time and the tree may fall easily.