Today's dogwood tree (Cornus florida) was known simply as "dog tree" in the 1500s; by 1614, the tree bore its common name of dogwood. The state tree of Missouri, dogwood grows throughout the eastern part of North America and bears white or pink flowers in the spring. Dogwood trees require only light pruning to remove unhealthy wood; gardeners should prune before the tree leafs out, in late winter to early spring. Pruning later in the spring could invite the dogwood borer, a serious pest.
Clip off suckers that grow along the trunk, below the graft site on your dogwood, or from the site of old pruning cuts. Suckers sap energy from the tree. They grow faster than regular wood and contain lots of foliage. The graft site should be visible as a horizontal scar near the base of your dog tree shrub.
Check over the branches of your shrub, looking for any that appear unhealthy. They may ooze sap, be discolored with a fungus or mildew or bear brown or black growths. They can also be bent, broken oh shriveled. Also note dead branches that don't move in the wind and feel hollow. All of this wood needs to be removed to keep your dog tree healthy.
Remove the unhealthy and dead wood you just identified, using lopping shears for branches thicker that 1 inch in diameter and hand pruners for smaller growth. After each cut, spray your tools with disinfectant so you don't accidentally spread disease to healthy parts of the tree.
Prune off branches that rub up against other branches, since this causes friction and will eventually damage the wood. Also prune away low growing branches that impede movement underneath the dog tree.