  We introduce a general recursive method to construct continuum random trees
(CRTs) from independent copies of a random string of beads, that is, any random
interval equipped with a random discrete probability measure, and from related
structures. We prove the existence of these CRTs as a new application of the
fixpoint method for recursive distribution equations formalised in high
generality by Aldous and Bandyopadhyay.
  We apply this recursive method to show the convergence to CRTs of various
tree growth processes. We note alternative constructions of existing
self-similar CRTs in the sense of Haas, Miermont and Stephenson, and we give
for the first time constructions of random compact R-trees that describe the
genealogies of Bertoin's self-similar growth fragmentations. In forthcoming
work, we develop further applications to embedding problems for CRTs, providing
a binary embedding of the stable line-breaking construction that solves an open
problem of Goldschmidt and Haas.
