$\nu (S)=\infty $, and set $$\begin{aligned}
\vartheta _{\mathrm{low}}:= &&\sup \left\{ b>0:\lim_{x\downarrow 0}x^{b}\nu
(s_{1}<1-x)=\infty \right\} , \\
\vartheta _{\mathrm{up}}:= &&\inf \left\{ b>0:\lim_{x\downarrow 0}x^{b}\nu
(s_{1}<1-x)=0\right\} .\end{aligned}$$ Then the height process $H_{F}$ is a.s. Hölder-continuous of order $\gamma $ for every $\gamma <\vartheta _{\mathrm{low}}\wedge |\alpha |$, and, provided that $\int_{S}(s_{1}^{-1}-1)\nu (\mathrm{d}\mathbf{s})<\infty ,$ a.s. not Hölder-continuous of order $\gamma $ for every $\gamma
>\vartheta _{\mathrm{up}}\wedge |\alpha |$.

Again we point out that one actually obtains an upper bound for the maximal Hölder coefficient even when $\int_{S}(s_{1}^{-1}-1)\nu (\mathrm{d}\mathbf{s})=\infty :$ with $\varrho $ defined by $\left( \ref{219}\right) ,$ a.s. $H_{F}$ cannot be Hölder-continuous of order $\gamma $ for any $\gamma >\vartheta _{\mathrm{up}}\wedge |\alpha |/\varrho .$

Note that $\vartheta _{\mathrm{low}},\vartheta _{\mathrm{up}}$ depend only on the characteristics of the fragmentation process, and more precisely, on the behavior of $\nu $ when $s_{1}$ is close to $1$. By contrast, our Hausdorff dimension result for the tree depended on a hypothesis on the behavior of $\nu $ when $s_{1}$ is near $0$. Remark also that $\vartheta _{\mathrm{up}}$ may be strictly smaller than $1$. Therefore, the Hausdorff dimension of $\mathcal{T}_{F}$ is in general not equal to the inverse of the maximal Hölder coefficient of the height process, as one could have 