Killing a Tree Manually
Killing a tree is hard work, particularly if you avoid using chemical assistance. You have to cut off a tree's water, food and/or sunlight at a critical time in it's life cycle to do the job. Herbicides work by gumming up or shutting down a tree's working parts to deprive the plant of one or more of the above. See: Chemical Weed-tree Control.
Using the Bark
Trees can be killed without herbicides or chemicals but extra time, patience, and understanding of tree anatomy is necessary. You most especially need to know about the function of a tree's inner bark - cambium, xylem and phloem - and how they combine forces to effect a tree's life. See: Parts of a Tree - Bark
The bark is a tree's most vulnerable body part above ground and the easiest target for an effective kill. Damaging enough roots to kill the tree quickly is complicated and hard to do without using chemicals.
Bark is made up of cork and phloem which protects the cambium and xylem. Dead xylem cells carry water and minerals from the roots to the leaves and is considered the tree's wood. Phloem, a living tissue, carries manufactured food (sugars) from the leaves to the roots. The cambium, which is a moist layer only a few cells thick, is the regenerative layer that gives birth to xylem on its inside and phloem to its outside.
Destroying the Bark
If the food-transporting phloem is severed all the way around the tree (a process called "girdling"), food cannot be carried to the roots and they will eventually die. As the roots die, so does the tree. Periods of rapid growth, usually from March until June in North America, are the best times to girdle a tree. These spring growth spurts are when tree bark "slips". The phloem and cork layer easily peels free, leaving the cambium and xylem exposed.
Remove as wide a section of bark as you have time for to make an adequate girdle ring. Then scrape (or chop) into the surface of the xylem to remove the cambium. If any cambial material remains, the tree will heal by overgrowing the girdle. The best time to girdle is before the trees leaf out. The process of leafing out will deplete energy stores from the roots, which stores cannot be renewed if the phloem conduit has been interrupted.
Avoid the Sprout!
Some trees are prolific sprouters and produce adventitious twigs near an injury. If you don't remove or kill the entire root, you just may have to control these sprouts. Sprouts coming out below the girdle must be removed as they will continue the process of feeding the roots if left to grow. When you are removing these sprouts, it is a good idea to check the girdled strip and remove any bark and cambium that may be trying to bridge the wound. Even cutting a tree down can't guarantee it will be killed. Many tree species, particularly some deciduous broad-leaf species, will sprout back from the original stump and root system.