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Recursive construction of continuum random trees and some contributions to the theory of tree-valued stochastic processes

We introduce a general recursive method to construct continuum random trees (CRTs) from i.i.d. copies of a string of beads, that is, any random interval equipped with a random discrete measure. We prove the existence of these CRTs as a new application of the fixpoint method formalised in high generality by Aldous and Bandyopadhyay. We further apply this recursive construction to "embed" Duquesne and Le Gall's stable tree into a binary compact CRT in a way that solves an open problem posed by Goldschmidt and Haas. Some of these developments are carried out in a space of ∞-marked metric spaces generalising Miermont’s notion of a k -marked metric space. Furthermore, we introduce a family of branch merging operations on continuum trees and show that Ford CRTs are distributionally invariant. This operation is new even in the special case of the Brownian CRT, which we explore in more detail. The operations are based on spinal decompositions and a regenerativity preserving merging procedure of (α, θ)-strings of beads, that is, regenerative strings of beads arising in the limit of ordered (α, θ)-Chinese restaurant processes as introduced by Pitman and Winkel. Indeed, we iterate the branch merging operation recursively and give a new approach to the leaf embedding problem on Ford CRTs related to (α, 2−α)-tree growth processes.