Alternatively, the class formula_11 admits a recursive representation formula_49 This gives two possibilities for formula_14.   where formula_53 stands for "draw a random variable formula_28; if the value formula_55 is returned, then execute formula_29 independently formula_57 times and return the sequence obtained". Here, formula_58 stands for the geometric distribution formula_59. As the first construction of the sequence operator suggests, Boltzmann samplers can be used recursively. If the target class formula_11 is a part of the system  where each of the expressions formula_62 involves only disjoint union, cartesian product and sequence operator, then the corresponding Boltzmann sampler is well defined. Given the argument value formula_12, the numerical values of the generating functions can be obtained by Newton iteration. Boltzmann sampling can be applied to labelled structures. For a labelled combinatorial class formula_11, exponential generating function is used instead:  where formula_8 denotes the number of labelled objects formula_4 of size formula_10. The operation of cartesian product and sequence need to be adjusted to take labelling into account, and the principle of construction remains the same. In the labelled case, the Boltzmann sampler for a labelled class formula_11 is required to output an object formula_4 with probability  In the labelled universe, a class formula_11 can be composed of all the finite sets of elements of a class formula_23 with order-consistent relabellings. In this case, the exponential generating function of the class formula_11 is written as  where formula_21 is the exponential generating function of the class formula_23. The Boltzmann sampler for formula_11 can be described as  where formula_80 stands for the standard Poisson distribution formula_81. In the cycle construction, a class formula_11 is composed of all the finite sequences of elements of a class formula_23, where two sequences are considered equivalent if they can be obtained by a cyclic shift. The exponential generating function of the class formula_11 is written as  where formula_21 is the exponential generating function of the class formula_23. The Boltzmann sampler for formula_11 can be described as  where formula_90 describes the log-law distribution formula_91. Let formula_92 denote the random size of the generated object from formula_14. Then, the size has the first and the second moment satisfying   The class formula_24 of binary trees can be defined by the recursive specification  and its generating function formula_22 satisfies an equation formula_100 and can be evaluated as a solution of the quadratic equation  The resulting Boltzmann sampler can be described recursively by  Consider various partitions of the set formula_103 into several non-empty classes, being disordered between themselves.