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