  We provide complete isomorphic invariance of a class of translation invariant
complemented subspaces of L_p constructed by Bourgain, Rosenthal and
Schechtman. We compute ordinal L_p-indices for this class. We further show that
the isometric index of a tree subspace over a well founded tree is an
invariance for the order of the tree. Finally we provide a dichotomy for the
subspaces of L_p with small ordinal indices.
