Cut the Ashe juniper tree shrubs at their trunk with a chainsaw or ax. Cut at the lowest point possible. Then cut up and burn the tree and its branches.
Cut a hole into the stump with your chainsaw or axe. Pour herbicide into the hole and then dig around the stump with a shovel to expose the roots. Sprinkle herbicide on the juniper's roots. Allow the herbicide to work as long as directed by the manufacturer. Additional doses may be required per the manufacturer's recommendations.
Cover the stump with a dark tarp and cinch it tight by wrapping a rope around it. This will deprive the stump of both rain water and sunlight while the herbicide kills the Ashe juniper stump and roots.