  We consider fixed-point equations for probability distributions on isometry
classes of measured metric spaces. The construction is required to be recursive
and tree-like, but we allow loops for the geodesics between points in the
support of the measure: one can think of a $\beta$-stable looptree decomposed
around one loop. We study existence and uniqueness of solutions together with
the attractiveness of the fixed-points by iterating. We obtain bounds on the
Hausdorff and upper Minkowski dimension, which appear to be tight for the
studied models. This setup applies to formerly studied structures as the
$\beta$-stable trees and looptrees, of which we give a new characterization and
recover the fractal dimensions.
