  In this article, the bi-Lipschitz embeddability of the sequence of countably
branching diamond graphs $(D_k^\omega)_{k\in\mathbb{N}}$ is investigated. In
particular it is shown that for every $\varepsilon>0$ and $k\in\mathbb{N}$,
$D_k^\omega$ embeds bi-Lipschiztly with distortion at most $6(1+\varepsilon)$
into any reflexive Banach space with an unconditional asymptotic structure that
does not admit an equivalent asymptotically uniformly convex norm. On the other
hand it is shown that the sequence $(D_k^\omega)_{k\in\mathbb{N}}$ does not
admit an equi-bi-Lipschitz embedding into any Banach space that has an
equivalent asymptotically midpoint uniformly convex norm. Combining these two
results one obtains a metric characterization in terms of graph preclusion of
the class of asymptotically uniformly convexifiable spaces, within the class of
separable reflexive Banach spaces with an unconditional asymptotic structure.
Applications to bi-Lipschitz embeddability into $L_p$-spaces and to some
problems in renorming theory are also discussed.
