Improved algorithms for ranking and unranking (k, m)-ary trees in B-order

Du and Liu (Eur J Comb 28:1312–1321, 2007) introduced (k, m)-ary trees as a generalization of k-ary trees. In a (k, m)-ary tree, every node on even level has degree k (i.e., has k children), and every node on odd level has degree m (which is called a crucial node) or is a leaf. In particular, a (k, m)-ary tree of order n has exactly n crucial nodes. Recently, Amani and Nowzari-Dalini (Bull Iranian Math Soc 45(4):1145–1158, 2019) presented a generation algorithm to produce all (k, m)-ary trees of order n in B-order using Zaks’ encoding, and showed that the generated ordering of this encoding results in a reverse-lexicographical ordering. They also proposed the corresponding ranking and unranking algorithms for (k, m)-ary trees according to such a generated ordering. These algorithms take $$\mathcal {O}(kmn^2)$$O(kmn2) time and space for building a precomputed table in which (k, m)-Catalan numbers (i.e., a kind of generalized Catalan numbers) are stored in advance. Then, each ranking and unranking algorithm can be performed subsequently in $$\mathcal {O}(n)$$O(n) and $$\mathcal {O}(n\log n)$$O(nlogn) time, respectively. In this paper, we revisit the ranking and unranking problems. With the help of an encoding scheme called “right-distance” introduced by Wu et al. (Math Comput Model 53:1331–1335, 2011a; IEICE Trans Inf Syst E94–D:226–232, 2011b), we propose new ranking and unranking algorithms for (k, m)-ary trees of order n in B-order using Zaks’ encoding. We show that both algorithms can be improved in $$\mathcal {O}(kmn)$$O(kmn) time and $$\mathcal {O}(n)$$O(n) space without building the precomputed table.