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.

我们考虑计算无穷二叉树的规则集合的度量的问题。虽然一般情况仍未解决，但我们证明，当以以下三种形式中的一种形式给出集合时，可以计算语言的度量：无后代关系的一阶公式；联合查询的布尔组合（具有后代关系）；或通过不确定的安全树自动机。此外，在前两种情况下，集合的度量始终是有理数，而在第三种情况下，它是代数数。此外，我们提供了使用后代关系并定义具有不合理（但代数）度量的无限树的语言的一阶公式的示例。