On a generalization of Bourgain's tree index

For a Banach space $X$, a sequence of Banach spaces $(Y_n)$, and a Banach space $Z$ with an unconditional basis, D. Alspach and B. Sari introduced a generalization of a Bourgain tree called a $(\oplus_n Y_n)_Z$-tree in $X$. These authors also prove that any separable Banach space admitting a $(\oplus_n Y_n)_Z$-tree with order $\omega_1$ admits a subspace isomorphic to $(\oplus_n Y_n)_Z$. In this paper we give two new proofs of this result.