Strip the bark off locust trees in summer to kill them. Insert a knife or sharp end of a crowbar into the lower portion of the tree trunk, a couple of feet off the ground. Pry a half-inch-thick piece of bark and wood off the trunk, removing a 6-inch-wide section. This method allows the hot sun to dry out the tree and eliminate the transport system carrying sugars from the leaves to the roots.
Leave the tree standing for one year, during which the roots will use up stored sugar and starve to death. Cut off new shoots as they grow because they will encourage the transportation of new sugars to the roots.
Cut the tree down to a stump that is about a foot from the ground. Cut shoots where they meet the stump with loppers or spray with an herbicide containing glyphosate to keep new locust trees from growing.