Many trees produce sucker growths from their root system, and if allowed to grow unchecked, they could in time, develop into trees.
If new or more stock of the tree producing the suckers is required consider propagating the suckers.(see here)
Suckers can appear from roots near the surface which have been damaged by digging / hoeing.
Among the most problematic trees are Cherries, Lilacs, Plums, and Poplars to name but a few.
The suckers should be removed as close as possible to their point of origin.
Do this by scraping away sufficient soil to reach the base of the sucker, then tear, rather than cut it away, so that most of the dormant basal buds are removed, thereby reducing the possibility of re-growth.
If sucker removal is difficult, sever the root to isolate the sucker from the parent tree, and then treat the wound with ammonium sulphamate.
This chemical may also kill surrounding plants if used in borders or near lawns.
When suckers are too numerous, removal of the tree may be the only solution.