  Starting from any graph on $\{1, \ldots, n\}$, consider the Markov chain
where at each time-step a uniformly chosen vertex is disconnected from all of
its neighbors and reconnected to another uniformly chosen vertex. This Markov
chain has a stationary distribution whose support is the set of non-empty
forests on $\{1, \ldots, n\}$. The random forest corresponding to this
stationary distribution has interesting connections with the uniform rooted
labeled tree and the uniform attachment tree. We fully characterize its degree
distribution, the distribution of its number of trees, and the limit
distribution of the size of a tree sampled uniformly. We also show that the
size of the largest tree is asymptotically $\alpha \log n$, where $\alpha = (1
- \log(e - 1))^{-1} \approx 2.18$, and that the degree of the most connected
vertex is asymptotically $\log n / \log\log n$.
