  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 $(\alpha, \theta)$-strings of beads, that is, random
intervals $[0, L_{\alpha, \theta}]$ equipped with a random discrete measure
$dL^{-1}$ arising in the limit of ordered $(\alpha, \theta)$-Chinese restaurant
processes as introduced recently by Pitman and Winkel. Indeed, we iterate the
branch merging operation recursively and give an alternative approach to the
leaf embedding problem on Ford CRTs related to $(\alpha,
2-\alpha)$-regenerative tree growth processes.
