If you really need to kill a tree, you can do it with a systemic herbicide like butoxyethyl ester. This is scary stuff because it gets taken up into the entire root, stem, and leaf system of a plant, and lingers for at least six months. It can cause colateral damage, contaminate soil and groundwater, and you can’t eat, compost, or burn anything that contains it. An alternative, for trees with a single trunk at least, is to cut the tree down to a stump and inject it with epsom salts.
Epsom salts is nothing more than magnesium sulphate, people use it in their baths to relax, and gardeners use it as a supplementary nutrient to rectify magnesium deficiencies in plants and trees. It’s also readily available, cheap and completely safe for people and the environment.
Large amounts of Epsom salts will draw moisture out of a stump much like an over-application of fertilizer does to roots, eventually drying it out, after which it will just naturally rot away. Any magnesium released into the soil will just be taken up by plants – magnesium is the key element in chlorophyll which allows plants to photosynthesize and makes leaves green.