We investigate notions of randomness in the space of nonempty closed subsets of the Cantor space. A probability measure is given and a version of the Martin-Löf test for randomness is deŽned. There are no random effectively closed sets but there are random closed sets in the next arithmetical level. It is shown that every random closed set is perfect, has measure 0, and has a certain box dimension. A random closed set has no n-c.e. elements. A closed subset of the Cantor space may be defined as the set of inŽnite paths through a tree and so the problem of compressibility of trees is explored, through prefix-free Kolmogorov complexity.