  Stick-breaking (SB) processes are often adopted in Bayesian mixture models
for generating mixing weights. When covariates influence the sizes of clusters,
SB mixtures are particularly convenient as they can leverage their connection
to binary regression to ease both the specification of covariate effects and
posterior computation. Existing SB models are typically constructed based on
continually breaking a single remaining piece of the unit stick. We view this
from a dyadic tree perspective in terms of a lopsided bifurcating tree that
extends only in one side. We show that several unsavory characteristics of SB
models are in fact largely due to this lopsided tree structure. We consider a
generalized class of SB models with alternative bifurcating tree structures and
examine the influence of the underlying tree topology on the resulting Bayesian
analysis in terms of prior assumptions, posterior uncertainty, and
computational effectiveness. In particular, we provide evidence that a balanced
tree topology, which corresponds to continually breaking all remaining pieces
of the unit stick, can resolve or mitigate several undesirable properties of SB
models that rely on a lopsided tree.
