We determine a class of rearrangements that admit a supporting tree. This
condition implies that the associated rearrangement operator has a bounded
vector valued extension. We show that there exists a large subspace of $L^p$ on
which a bounded rearrangement operator acts as an isomorphism. The
combinatorial issues of these problems give rise to a two-person game, to be
played with colored dyadic intervals. We determine winning strategies for each
of the players.