Not sure where you came up with the notion that mulch harms tree roots - it most certainly does not! Mulch is very beneficial to trees, especially young ones but ALL trees can benefit from a good quality mulch. It helps to prevent weeds from developing or lawn grasses growing right up to the tree base, which can rob them of nutrients and soil moisture and complicates garden maintenance. As it decomposes, it adds nutrients to the soil. Mulch also helps to retain soil moisture and insulates against swings in soil temperatures. All of these are good things :-) The only thing you want to avoid is piling mulch up against the tree trunk - keep it a good several inches away from the wood.

If you prefer not to use mulch, planting the ajuga or any other suitable ground cover is fine.