  We introduce a random finite rooted tree $\mathcal{C}$, the steady state
cluster, characterized by a recursive description: $\mathcal{C}$ is a singleton
with probability $1/2$ and otherwise is obtained by joining by an edge the
roots of two independent trees $\mathcal{C}'$ and $\mathcal{C}''$, each having
the law of $\mathcal{C}$, then re-rooting the resulting tree at a uniform
random vertex.
  We construct a stationary regenerative stochastic process $\mathcal{C}(t)$,
the steady state cluster growth process. It is characterized by a simple
fixed-point property. Its stationary distribution is the law of the steady
state cluster $\mathcal{C}$. We conjecture that $\mathcal{C}(t)$ is the local
limit of the evolution of the cluster of a tagged vertex in the stationary
state of the mean field forest fire model of R\'ath and T\'oth. We describe its
explosions in terms of a L\'evy subordinator, using a state-dependent time
change.
  The steady state cluster is also a multitype Galton-Watson tree with a
continuum of types. The steady state cluster conditioned on its size is a
random weighted spanning tree of the complete graph equipped random edge
weights with a simple explicit joint distribution. The time-reversal of the
steady state cluster growth process is realised as the component of a `uniform'
vertex in a logging process of a critical multitype Galton-Watson tree
conditioned to be infinite.
  We construct a stationary forest fire model on the infinite rooted tree
$\mathbb{Z}^*$ with the property that the evolution of the cluster of the root
is a version of the steady state cluster growth process. This model is similar
in spirit to Aldous' frozen percolation model on the rooted infinite binary
tree. We conjecture that it is the local weak limit of the stationary
R\'ath-T\'oth model.
