The Order of the Non-universal Tree of a Hilbert Space with Respect to the Haar Basis

In his 1994 doctoral thesis, Bossard introduced the notion of the non-universal tree $T_{NU}(X)$ associated with each separable Banach space $X$ which does not contain an isomorphic copy of $C(2^{\mathbb{N}})$. Together with the order operation defined on well-founded trees, we obtain a method of classifying the complexity of separable Banach spaces by the degree of isomorphism of finite-dimensional subspaces of $C(2^{\mathbb{N}})$. Despite further refinement of this concept in later years, we are unaware of any specific instances of direct exhibitions of the order of the non-universal tree for a concrete space and basis. We show that if $H$ is any separable Hilbert space, then $o(T_{NU}(H))=\omega +1$ when taken with respect to the Haar basis for $C(2^{\mathbb{N}})$, demonstrating that the class of Hilbert spaces is the least complex class with respect to this measurement when considering this basis.