  Trees in Brownian excursions have been studied since the late 1980s. Forests
in excursions of Brownian motion above its past minimum are a natural extension
of this notion. In this paper we study a forest-valued Markov process which
describes the growth of the Brownian forest. The key result is a composition
rule for binary Galton--Watson forests with i.i.d. exponential branch lengths.
We give elementary proofs of this composition rule and explain how it is
intimately linked with Williams' decomposition for Brownian motion with drift.
