We study a notion of shuffle for trees which extends the usual notion of a shuffle for two natural numbers. Our notion of shuffle is motivated by the theory of operads and occurs in the theory of dendroidal sets. We give several equivalent descriptions of the shuffles, and prove some algebraic and combinatorial properties. In addition, we characterize shuffles in terms of open sets in a topological space associated to a pair of trees. This is a joint work with Ieke Moerdijk.[-]