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, random intervals $[0, L_{α, θ}]$ equipped with a random discrete measure $dL^{-1}$ arising in the limit of ordered $(α, θ)$-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 $(α, 2-α)$-regenerative tree growth processes.