Mulch the area under the tree that will not grow grass. Mulching is one of the best things you can do for your plants. It slows weeds and diseases, insulates their roots from heat and cold and reduces water loss from the soil surface. If the area is poorly drained already, do not much though. It may keep the area too wet.
Mulches are a blanket of protection for plants.
You can water and fertilize right through them. I do not recommend stone mulches. They get hot in the summer and can increase water loss. They may be useful in shady areas where there are no trees to litter the mulch, but these areas are rare. Another cover-up may be required on plants that cannot take the cold.