Many species of trees have epicormic buds under the bark of the tree that normally lie dormant. But when a tree is under stress, the buds send out shoots (also known as suckers or water sprouts) in an attempt to recover. Such shoots can be caused by pruning, insects, disease, fire, or damage to a part of the tree. Some trees can regenerate after stress by means of those sprouts, but many never fully recover.
Epicormic shoots are one of the most obvious signs of emerald ash borer infestations on ash trees. As the top of the tree is attacked, the tree tries to continue living by pushing up sprouts below the insect activity.
If your non-ash tree seems otherwise healthy, remove the shoots to maintain the tree’s form. Make your cut as close to the base of the shoot as possible.
If it’s an ash tree and 30% of the canopy is dead, the tree will probably not survive. See https://extension.entm.purdue.edu/EAB/ for more information.
In the craziest example we’ve seen, several West Lafayette Tree Friend members made a heroic effort to get this ginkgo tree on Vine Street back in shape. They cut back a forest of shoots from the base. Not perfect, but much better.