(Courtesy of US Department of Agriculture)
Proper mulching and pruning of trees are important to their health. The following information from the USDA Forest Service addresses these two issues.
Maintain a ring of mulch around the tree (the wider the better). Organic material like wood chips and leaves are best. Wood chips will take longer to break down and, therefore, will not require replacement as often.
Support the branch with one hand while you make the cut to prevent the bark from ripping. If the branch is too large to support, use the three-step method (see details below). For the final cut, look for the branch bark ridge and trunk collar. Begin the cut just outside of the branch bark ridge, and angle down away from the trunk. Stay close to the trunk collar without cutting into it (see images below)