Ask Mr. Smarty Plants

The Pride  of Houston Yaupon is a cultivar of Ilex vomitoria (yaupon) that is noted for its abundant production of red berries. Another trait that is not so endearing is the production of sprouts from the roots that help the plant spread in its native habitat.

The paragraph below is an excerpt from Fact Sheet ST-311, "Ilex vomitoria Yaupon Holly", published by the US Forest Service, and addresses the sprouting problem.

USE AND MANAGEMENT 
A tough native of the southern United States, Yaupon Holly grows quickly in a variety of locations, from full sun or shade to seaside or swamps, in sand or clay. Crowns will be thin in the shade.

It will grow in soil with a pH in the 7’s and is very tolerant of drought and sea salt, It sprouts readily from the
roots forming clumps of upright shoots beneath the canopy. Sprouting is most troublesome if the soil beneath the canopy is disturbed, as in planting shrubs or flowers under the tree.
 These need to be pruned to the ground two or three times each year to maintain a neat appearance.

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At least your situation doesn't seem as dire as the one addressed in this answer to a previous question.