  In this paper, we investigate compact ultrametric measure spaces which form a
subset $\mathcal{U}^w$ of the collection of all metric measure spaces
$\mathcal{M}^w$. Similar as for the ultrametric Gromov-Hausdorff distance on
the collection of ultrametric spaces $\mathcal{U}$, we define ultrametric
versions of two metrics on $\mathcal{U}^w$, namely of Sturm's distance of order
$p$ and of the Gromov-Wasserstein distance of order $p$. We study the basic
topological and geometric properties of these distances as well as their
relation and derive for $p=\infty$ a polynomial time algorithm for their
calculation. Further, several lower bounds for both distances are derived and
some of our results are generalized to the case of finite ultra-dissimilarity
spaces.
