Regular n-ary queries in trees are queries which are definable by an MSO formula with n free first-order variables. We investigate the variable independence problem -- originally introduced for databases -- in the context of trees. In particular, we show how to decide whether a regular query is equivalent to a union of cartesian products, independently of the input tree. As an intermediate step, we reduce this problem to the problem of deciding whether the number of answers to a regular query is bounded by some constant, independently of the input tree. As a (non-trivial) generalization, we introduce variable independence w.r.t. a dependence forest between blocks of variables, which we prove to be decidable.