How do I treat a dogwood tree displaying brown spots on leaves ?
Most of the time the brown spots on dogwoods is Spot Anthracnose. The disease is unsightly, but typically not very harmful to the tree. A fungicide treatment for this disease is normally not warranted. You should remove any infected leaves, including dead material on the tree, and compost the material thoroughly or dispose of it to stop the disease life-cycle.
However, there are a couple of other possibilities: Dogwood Anthracnose, Powdery Mildew or Leaf Scorch.
Please refer to these two links for symptoms of the diseases and control:
If you suspect dogwood anthracnose, get it diagnosed by a professional prior to a fungicide application. The University of Missouri Plant Diagnostic Clinic can be of help with diagnosing dogwood anthracnose - http://plantclinic.missouri.edu/