Regular n-ary queries in trees and variable independence

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.