  We consider a graph coloring algorithm that processes vertices in order taken
uniformly at random and assigns colors to them using First-Fit strategy. We
show that this algorithm uses, in expectation, at most $(1 + o(1))\cdot \ln n
\,/\, \ln\ln n$ different colors to color any forest with $n$ vertices. We also
construct a family of forests that shows that this bound is best possible.
