The usual method of tree removal, cutting off a tree near the ground, leaves a stump that has to be removed later. It is much simpler, and usually faster, to remove the stump along with the tree because the tree can be used as a lever to pry the stump out of the ground.
For a small tree, simply use a shovel to cut a circle as deep as the shovel blade around the trunk. This is even easier if the shovel edge is sharp and the soil is wet. It may be necessary to stomp or jump on the shovel to make it cut through the roots. Once a circle has been cut around the tree, pushing on the tree will usually break the remaining roots so the entire tree can be lifted out.
If the roots are too thick to cut with a shovel, the next option is to dig a trench around the tree about a foot out from the trunk and a foot deep to expose the roots. The roots can be cut with an axe or a saw.
A better method is to use a reciprocating saw, such as a Sawzall, to cut the circle and the roots. Hardened blades are available for reciprocating saws which will cut through soil and roots, although the saw blade gets dull after a several minutes and needs changing. Rocky ground dulls the blade quickly.
For a medium to large tree, first remove limbs one at a time. To protect plants or objects under the tree, large limbs should be tied with a rope, cut and then lowered to the ground. Leave a eight foot tall trunk to provide enough leverage to break the remaining roots.
One drawback to this method is that it does not provide a good way to guide the tree as it falls. Medium to large tree need to have the main branches and top removed first.
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