Pachysandra aka Japanese Spurge will crowd out any and all weeds, and they do well under trees as long as there is ample depth to the soil so they can spread with underground runners. Roots aren't very deep.

Very low maintenance and evergreen, weeds will never get through the root system once they're established.

Otherwise, a very low ground creeping groundcover is Mazus Reptans, but they need full sun. LOVE the stuff.

My favorites are evergreen european ginger, Asarum europaeum, and Epimediums. They both seem to crowd out weeds. Both appreciate shade and dappled light. I considered them pricey and they increase slowly. I must string trim the Epimedium down, once a year, in Feb. as they get really bad looking. This also allows for a more easily seen blooming.