expected. However, this turns out to be true in the case of the stable tree, as will be checked in Section \[heightstable\]:

\[C2\] The height process of the stable tree with index $\beta \in (1,2]$ is a.s. Hölder-continuous of any order $\gamma <1-1/\beta $, but a.s.not of order $\gamma >1-1/\beta $.

When $\beta =2$, this just states that the Brownian excursion is Hölder-continuous of any order $<1/2$, a result that is well-known for Brownian motion and which readily transfers to the normalized Brownian excursion (e.g. by rescaling the first excursion of Brownian motion whose duration is greater than $1$). The general result had been obtained in [@duqleg02] by completely different methods.

Last, we mention that most of our results extend to a more general class of fragmentations in which a fragment with mass $x$ splits to give fragments with masses $x$**s**$,$ $\mathbf{s}\in S,$ at rate $\varsigma (x)\nu ($d$\mathbf{s})$ for some non-negative continuous function $\varsigma $ on $\left( 0,1\right] $ (see [@haas02] for a rigorous definition). The proofs of the above theorems easily adapt to give the following results: when $\liminf_{x\rightarrow 0}x^{-b}\varsigma (x)>0$ for some $b<0,$ the fragmentation can be encoded as above into a CRT and, provided that $\nu $ 