  We consider an infinite homogeneous tree $\mathcal V$ endowed with the usual
metric $d$ defined on graphs and a weighted measure $\mu$. The metric measure
space $(\mathcal V,d,\mu)$ is nondoubling and of exponential growth, hence the
classical theory of Hardy and $BMO$ spaces does not apply in this setting. We
introduce a space $BMO(\mu)$ on $(\mathcal V,d,\mu)$ and investigate some of
its properties. We prove in particular that $BMO(\mu)$ can be identified with
the dual of a Hardy space $H^1(\mu)$ introduced in a previous work and we
investigate the sharp maximal function related with $BMO(\mu)$.
