Banach space, properties which might not be determined by the finite dimensional subspaces. A result in this direction was obtained by Baudier, Kalton and Lancien in [@BKL]. They showed that a reflexive Banach space $X$ has a renorming, which is [*asymptoticaly uniformly convex*]{} (AUC) and [*asymptoticaly uniformly smooth*]{} (AUS), if and only if the countably branching trees of length $n\in{\mathbb{N}}$, $T_n$ do not uniformly bi-Lipschitzly into $X$. Here $T_n=\bigcup_{k=0}^n{\mathbb{N}}^k$, together with the graph metric, [*i.e.,* ]{}$d( a, b)= i+j-\max\{ t\ge 0: a_s=b_s, s=1,2\ldots t\} $, for $a=(a_1,a_2,\ldots a_i)\not=  b=(b_1,b_2,\ldots b_j)$ in $T^n$. Among the many equivalent conditions for a reflexive Banach space $X$ to be (AUC)- and (AUS)-renormable (see [@OS2]) one of them states that ${\text{\rm Sz}}(X)={\text{\rm Sz}}(X^*)=\omega$, where ${\text{\rm Sz}}(Z)$ denotes the [*Szlenk index*]{} of a Banach space $Z$ (see Section \[S:6\] for the definition and properties of the Szlenk index). In [@DKLR] Dilworth, Kutzarova, Lancien and Randrianarivony, showed that a separable Banach space $X$ is reflexive and (AUC)- and (AUS)-renormable if and only $X$ admits an equivalent norm for which $X$ has [*Rolewicz’s $\beta$-property*]{}. According to [@Ku1] a Banach space $X$ has Rolewicz’s $\beta$-property if and only if $$\bar \beta_X(t)= 1-\sup\Big\{ \inf\Big\{\frac{\|x- x_n\|}{2}: n\ge1\Big\}: (x_n)_{n=1}^\infty\subset B_X,\,\, \text{sep} \big[(x_n)\big]\ge 