There's a quick fix for a tree that chokes out the sunlight, and with it any hope of growing grass: Encircle the tree with shade plants.
Although plantings around a tree create a lovely landscape, you need to observe a few precautions to protect the tree. To avoid severing or damaging tree roots, do not dig under a tree's canopy. Instead of digging, spread a layer of topsoil no more than 4 inches deep over the ground. Cover this layer with a 2-inch-deep mulch of chopped leaves, which will decompose into a rich humus. Dig your planting holes in this new medium. Another precaution -- when planting, be careful not to damage the tree trunk.
The photograph shows a circle of Hostas and Impatiens, two staples of a shade garden. You also can jazz up the areas beneath trees with other shade-loving plant: