  We give a new rapid mixing result for a natural random walk on the
independent sets of a graph $G$. We show that when $G$ has bounded treewidth,
this random walk -- known as the Glauber dynamics for the hardcore model --
mixes rapidly for all fixed values of the standard parameter $\lambda > 0$,
giving a simple alternative to existing sampling algorithms for these
structures. We also show rapid mixing for analogous Markov chains on dominating
sets, $b$-edge covers, $b$-matchings, maximal independent sets, and maximal
$b$-matchings. (For $b$-matchings, maximal independent sets, and maximal
$b$-matchings we also require bounded degree.) Our results imply simpler
alternatives to known algorithms for the sampling and approximate counting
problems in these graphs. We prove our results by applying a divide-and-conquer
framework we developed in a previous paper, as an alternative to the
projection-restriction technique introduced by Jerrum, Son, Tetali, and Vigoda.
We extend this prior framework to handle chains for which the application of
that framework is not straightforward, strengthening existing results by Dyer,
Goldberg, and Jerrum and by Heinrich for the Glauber dynamics on $q$-colorings
of graphs of bounded treewidth and bounded degree.
