Hi, Rebecca!  Sorry for the late response.  A great way to keep those shoots from coming back is to use a selective chemical herbicide that will kill the shoots without damaging the tree.  This chemical can be found at local grocery stores and nurseries.  If you prefer a non-chemical means of control, you could also mow these suckers down.  Let us know if you have any other questions!

Hi Rebecca!

The active ingredient is Ethyl 1-Naphthaleneacetate and the product is typically referred to as a "sucker punch".  Once applied, it should last up to 6 months.