  We use a natural ordered extension of the Chinese Restaurant Process to grow
a two-parameter family of binary self-similar continuum fragmentation trees. We
provide an explicit embedding of Ford's sequence of alpha model trees in the
continuum tree which we identified in a previous article as a distributional
scaling limit of Ford's trees. In general, the Markov branching trees induced
by the two-parameter growth rule are not sampling consistent, so the existence
of compact limiting trees cannot be deduced from previous work on the sampling
consistent case. We develop here a new approach to establish such limits, based
on regenerative interval partitions and the urn-model description of sampling
from Dirichlet random distributions.
