  We consider fixed-point equations for probability measures charging measured
compact metric spaces that naturally yield continuum random trees. On the one
hand, we study the existence/uniqueness of the fixed-points and the convergence
of the corresponding iterative schemes. On the other hand, we study the
geometric properties of the random measured real trees that are fixed-points,
in particular their fractal properties. We obtain bounds on the Minkowski and
Hausdorff dimension, that are proved tight in a number of applications,
including the very classical continuum random tree, but also for the dual trees
of random recursive triangulations of the disk introduced by Curien and Le Gall
[Ann Probab, vol. 39, 2011]. The method happens to be especially efficient to
treat cases for which the mass measure on the real tree induced by natural
encodings only provides weak estimates on the Hausdorff dimension.
