A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local ``random-to-front'' reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the transition matrix are determined using Brown's theory of random walk on semigroups.
"Note: Random-to-front shuffles on trees." Electron. Commun. Probab. 14 36 - 41, 2009. https://doi.org/10.1214/ECP.v14-1445