Some trees indeed are allelopathic, meaning they produce a chemical that inhibits growth of another species (nothing grows under a Black Walnut, for example), but it looks like Liriope is growing under the first tree. Under the other tree, it's hard to see, but it might be Dwarf Red Ginger on the right hand side, and in the circular area around the bottom is more Liriope.

I believe that the dark green "hairy" looking plants under the closer tree are mondograss or "monkey" grass.