Is there a ground cover plant that is tree friendly and weed "unfriendly?"?

Hi! I have a 100 ft. by 6 ft. strip in my back yard that has a line of young (8 to 10 ft.) Leeland Cypress trees and a mulch fill. We weed this thing like crazy. Even with landscape fabric under the mulch, we still get some crazy stuff growing back there. I was wondering… is there a ground cover plant that we could put in there that would push out (or maybe just hide) the weeds that would look good under the trees that also wouldn’t harm them? Thanks!