You have two good choices for the best weed barrier—organic mulch or high-quality landscape fabric. (Don't use black plastic. It doesn't allow rain to reach plant roots, and it traps water vapor, which facilitates the growth of mold and mildew.)
An organic mulch such as shredded bark, in a layer several inches deep, will help control weeds. It will also decompose over time, which will add organic matter to the soil and make your tree happier. However, you'll have to add more organic mulch every couple of years. Landscape fabric, when protected from sunlight, decomposes more slowly than organic mulch and doesn't need to be replaced as often. However, the type of landscape fabric makes a difference. Hold the fabric up to the light and make sure the pores in the fabric are small enough to prevent weeds from growing through the barrier. Also, a good-quality landscape fabric is one you can't tear or stretch easily. It should feel stiff, not flimsy and limp.