Starting from any graph on {1, ..., 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, ..., 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 α log n, where α = (1 − log(e − 1)) −1 ≈ 2.18, and that the degree of the most connected vertex is asymptotically log n/ log log n.