We urge you to have mulch around your trees in preference to grass or just soil. Whilst grass can have deeper roots than trees, effectively out competing them for moisture and nutrients, this is not the main reason for this.
Mulch will increase Mycorrhizal populations by up to 15 times that of grass. This leads to a much healthier tree.
Mulching is the closest a tree will get to the floor covering of a forest. In a forest the recycling of organic matter, feeds the soil, which provides the roots and their fungi with their nutrients. On a much smaller scale your mulching will also assist.