HI Tom, thanks for the question.  

If you are having a problem with Hawthorn other than pruning back suckers, you are probably dealing with Common Hawthorn (Crataegus monogyna).   This plant is spread by the berries and birds.  Check out the following link which details control methods for this plant:

Common Hawthorn | solvepest (oregonstate.edu)

I hope this is helpful.

It's possible I may not know the type of tree I'm dealing with, so I'm sending pictures. It sends up tons of baby trees from the root system.

Hi Tom,

The tree in the picture is an American Plum (Prunus Americana).  The only sure way is to kill it and all the suckers with a systemic herbicide.  Other than that, removing suckers at their base is the thing to do.  Check out the following link:

Managing Suckers on Fruit, Ornamental, and Shade Trees | Horticulture and Home Pest News (iastate.edu)