The uniform measure of simple regular sets of infinite trees

We consider the problem of computing measure a regular set infinite binary trees. While general case remains unsolved, we show that language can be computed when is given in one following three formalisms: first-order formula with no descendant relation; Boolean combination conjunctive queries (with relation); or by non-deterministic safety tree automaton. Additionally, first two cases always rational, while third it an algebraic number. Moreover, provide example uses relation and defines trees having irrational (but algebraic) measure.