A solid shoetree is a shoetree with two pieces of wood (or plastic, depending on the material used). The first piece is the heel which is attached to a metal rod, and the second piece, also attached to that metal rod, is for the toe area. The signature aspect of a solid shoetree is that the toe portion is not two pieces, one on either side of the rod. That is the split toe shoetree.
Why you want it: Great for rigid shoes. Not so good for soft leather.