Answer: Suckers growing from the base of a tree can be very annoying. One way to get the upper hand is to yank the suckers off rather than cutting them off. By pulling the suckers off you'll be removing the cells and tissues that encourage regrowth. Cutting the suckers to ground level has the opposite effect and will encourage further production. Suckers that grow from the roots will just have to be cut off. There's nothing else that you can do that won't harm the tree. Don't treat the area with any chemicals or you'll compromise the health of your tree. (Anything you apply will be transported throughout the vascular system of the tree.) Just keep after those suckers, removing them before they get too big.
Grass always has a hard time growing beneath trees. Shade is one factor, dry soil is the other. The roots of trees drink up all the moisture and the grass roots simply cannot compete. However, if you water more frequently in the shady areas under the tree and if you choose a shade tolerant grass type, you should be able to maintain a green lawn for most of the year. However - most shade tolerant grasses are cool season grasses and will die out during warm weather (July and August). So I suspect you already have cool season grasses growing beneath your tree. If you want to reseed, use fine fescue.
The most common turf-type fine fescues include creeping red fescue (Festuca rubra), Chewings fescue (Festuca rubra var. commutata), hard fescue (Festuca longifolia), and sheep fescue (Festuca ovina). These species are used extensively for lawns, grounds, and parks in your part of Virginia. They are ideal for low maintenance turfs, but, are not typically used for sports turfs. During cool weather (and when properly maintained), the fine fescues produce an attractive, uniform stand with a medium-green to dark-green color. These grasses are extremely fine-textured and are compatible in mixtures of most cool-season turfgrasses. As a group, the fine fescues tolerate soils of low fertility and low pH, droughty soils, and shaded conditions. They are not well adapted to hot, humid conditions; poorly drained soils; high-traffic areas such as athletic fields or playgrounds; and high rates of nitrogen fertilizer. Like Kentucky bluegrass, the fine fescues become semi-dormant under prolonged periods of heat and drought and recover quickly with the advent of cooler temperatures and adequate soil moisture.
Best wishes with your landscape!
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