A Constant Amortized Time Algorithm for Generating Left-Child Sequences in Lexicographic Order

Wu et al. (Theoret. Comput. Sci. 556:25–33, 2014) recently introduced a new type of sequences, called left-child sequences (LC-sequences for short), for representing binary trees. They pointed out that such sequences have a natural interpretation from the view point of data structure and gave a characterization of them. Based on this characterization, Pai et al. (International conference on combinatorial optimization and applications. Springer, Cham, pp. 505–518, 2016) showed that there is an easily implementing algorithm that uses generate-and-test approach to filter all LC-sequences of binary trees with n internal nodes in lexicographic order, while in general this algorithm is not efficient at all. In this paper, we design two novel rotations that allow us to drastically alter the shape of binary trees (and thus their corresponding LC-sequences). As an application, these operations can be employed to generate all LC-sequences in lexicographic order. Accordingly, we present a more efficient algorithm associated with the new types of rotations for generating all LC-sequences and show that it takes only constant amortized running cost.