Here are a few suggestions: Snow-on-the-Mountain (Aegopodium podagraria 'Variegatum'), Lily-of-the-Valley (Convallaria majalis), Pachysandra (P. terminalis), and Ajuga (A. reptans). All of these groundcovers are aggressive enough to compete with the willow tree roots. You will have to remove the straggly bits of turf under the tree. Installing a finished edge between the groundcover and the lawn is optional. Regular lawn mowing will help keep the groundcovers in bounds, but the boundary will get blurred without a defined edge. For a longer list of groundcover suggestions, try a detailed search with the Plant Finder on BHG.com.