  For $0<\alpha<1,$ and $\theta>-\alpha,$ let
$(S^{-\alpha}_{\alpha,\theta+r})_{\{r\ge 0\}}$ denote an increasing(decreasing)
sequence of variables forming a time inhomogeneous Markov chain whose marginal
distributions are equivalent to generalized Mittag Leffler distributions. We
exploit the property that such a sequence may be connected with the two
parameter $(\alpha,\theta)$ family of Poisson Dirichlet distributions. We
demonstrate that the sequences serve as limits in certain types of preferential
attachment models. As one illustrative application, we describe the explicit
joint limiting distribution of scaled degree sequences arising under a class of
linear weighted preferential attachment models as treated in M\'ori (2005),
with weight $\beta>-1.$ When $\beta=0$ this corresponds to the Barbasi-Albert
preferential attachment model. We then construct sequences of nested
$(\alpha,\theta)$ Chinese restaurant partitions of $[n]$. From this, we
identify and analyze relevant quantities that may be thought of as mimics for
vectors of degree sequences, or differences in tree lengths. We also describe
connections to a wide class of continuous time coalescent processes that can be
seen as a variation of stochastic flows of bridges related to generalized
Fleming-Viot models. Under a change of measure our results suggest the
possibilities for identification of limiting distributions related to
consistent families of nested Gibbs partitions of $[n]$ that would otherwise be
difficult by methods using moments or Laplace transforms. In this regard, we
focus on special simplifications obtained in the case of $\alpha=1/2.$ That is
to say, limits derived from a $\mathrm{PD}(1/2|t)$ distribution. Throughout we
present some distributional results that are relevant to various settings. We
describe nestings across the $\alpha$ parameter in section 6
