Rooted, weighted continuum random trees are used to describe limits of
sequences of random discrete trees. Formally, they are random quadruples
$(\mathcal{T},d,r,p)$, where $(\mathcal{T},d)$ is a tree-like metric space,
$r\in\mathcal{T}$ is a distinguished root, and $p$ is a probability measure on
this space. The underlying branching structure is carried implicitly in the
metric $d$. We explore various ways of describing the interaction between
branching structure and mass in $(\mathcal{T},d,r,p)$ in a way that depends on
$d$ only by way of this branching structure. We introduce a notion of
mass-structure equivalence and show that two rooted, weighted
$\mathbb{R}$-trees are equivalent in this sense if and only if the discrete
hierarchies derived by i.i.d. sampling from their weights, in a manner
analogous to Kingman's paintbox, have the same distribution. We introduce a
family of trees, called "interval partition trees" that serve as
representatives of mass-structure equivalence classes, and which naturally
represent the laws of the aforementioned hierarchies.