  Consider the Aldous Markov chain on the space of rooted binary trees with $n$
labeled leaves in which at each transition a uniform random leaf is deleted and
reattached to a uniform random edge. Now, fix $1\le k < n$ and project the leaf
mass onto the subtree spanned by the first $k$ leaves. This yields a binary
tree with edge weights that we call a "decorated $k$-tree with total mass $n$."
We introduce label swapping dynamics for the Aldous chain so that, when it runs
in stationarity, the decorated $k$-trees evolve as Markov chains themselves,
and are projectively consistent over $k\le n$. The construction of projectively
consistent chains is a crucial step in the construction of the Aldous diffusion
on continuum trees by the present authors, which is the $n\rightarrow \infty$
continuum analogue of the Aldous chain and will be taken up elsewhere. Some of
our results have been generalized to Ford's alpha model trees.
