Use products with the active ingredient glyphosate to kill the tree, suggests the Washington State University Extension. Buy the undiluted, water-soluble formula, because this is more effective than esters.
Using a sharp axe, cut the tree down to a stump. Make sure the surface is smooth and level to keep the herbicide from flowing off.
If the stump diameter is less than 3 inches, spray herbicide on the entire stump surface. If stumps are greater than 3 inches in diameter, spray only the 2 to 3 inches right next to the bark. The heartwood of larger trees is already dead.
Use herbicide on the freshly cut stump right away for best effect. If you are not able to do this immediately, make a new cut to expose fresh tree tissue before you use the herbicide. The best time to use glyphosate is during late spring and early summer.
Remove the dead stump by grinding it with a stump grinder that you can rent at a garden supply store. The machine is like a meat grinder that has a rotating wheel with carbide teeth that disintegrates the stump into chips. Grind to about a foot below the soil line.