Sue Says
As you are finding out, many trees and shrubs send up shoots from their roots and around the base of the plant. These are called suckers, and can be rather inconvenient when they appear in lawns and borders.
The best way to remove suckers is to scrape back the soil to the base of the sucker and remove it by either cutting or tearing it away from the root. Tearing it is preferential as this will remove some of the basal buds and reduce the risk of regrowth. Where you are tackling larger suckers it may be necessary to sever the root of the parent plant to separate it, before applying a glyophosate based herbicide to the foliage of the sucker in order to kill it. Never apply glyphosate to the sucker when it is still attached to the parent tree as this may kill or damage the parent too.