Select an herbicide with the active ingredient glyphosate to effectively kill the stink tree, suggests the Washington State University Extension. Choose an undiluted, water soluble formula as this has higher efficacy as compared to esters.
Cut the tree down to a stump using a sharp axe. Make sure that the stump surface is smooth and level to keep the chemical from sliding off.
Use the herbicide on the fresh cut stump right after cutting. If you were not able to do this immediately, make a new cut on the stump to expose fresh wood prior to using chemical.
Spray the herbicide over the entire surface of the stump if the stump diameter is less than 3 inches. If the stump has a diameter of over 3 inches, spray only the 2 to 3 inches right next to the bark as the heartwood of larger trees is already dead.
Avoid using glyphosate on very hot days or on extremely drought stressed trees. The recommended time to use glyphosate is during August and September, cites the Washington State University Extension.
Let the stump stay in place or use a stump grinder to grind to a depth of about a foot below ground level.