What is this tree? It sends out suckers everywhere.

This is Black Tupelo, also called blackgum or sour gum. These trees are common in the Mid-South and produce many suckers from the roots. Retired UT Extension Agent Mike Dennison says the best way to remove the suckers is to cut them off (he suggests a blade on a weed eater). Mike does not suggest using herbicides because you run the risk of killing the whole tree.
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Related Resources:
Nyssa Sylvatica
Black Gum, Sour Gum, Black Tupelo: Nyssa sylvatica