Tree sucker sprouts and watersprouts are vigorous, upright, epicormic shoots that grow from dormant buds on older wood. They are mostly a problem on fruit and landscape trees, can grow very large in one season and occur most often under stressful conditions like drought, after severe pruning and limb loss.
Sucker sprouts and water sprouts can provide clues to a tree's health. Both types of sprouts often indicate vigorous growth below the sprouts but may suggest that the tree has injury or dead wood above the sprouts. The tree is trying to compensate using these sprouts to increase vigor.
Watersprouts and suckers differ primarily in their location on the tree. Watersprouts arise above the trunk graft while suckers arise below the graft union on the original grafting stock's trunk and roots. Both kinds of sprouts should be removed immediately while remembering that a watersprout can be developed into the main trunk if there is severe damage above it. Watersprouts are easily pulled off.
Basal and root suckers should always be removed. They are usually found at the tree base but can also sprout from roots several feet away from the trunk. Suckers should be removed at the root or tree base connection. Tear rather than cut away the shoot so that the majority of dormant basal buds are removed thus reducing the possibility of re-growth.
Some sources suggest suggest applying glyphosate or triclopyr to the sprouts but only if you severe the root from the tree. My major concern here would be the very real possibility of herbicide damage or harmful root loss. Do this understanding that there is risk of harming the tree!
Removal of the tree may be your only solution when suckers are too numerous. You will then need to apply a brush killer to control the sprouts.