Ranking and Unranking of t-ary Trees Using the Right Distance Representation

In this paper, we introduce a concise representation, called right-distance sequences (or RD-sequences for short), to describe all t-ary trees with n internal nodes. A result reveals that there exists a close relationship between the representation and the well-formated integer sequences suggested by Zaks in 1980. Using recursion tree and its concomitant tables, a systematical way can help us to investigate the structural representation of t-ary trees. Consequently, we develop efficient algorithms for determining the rank of a given t-ary tree (i.e., ranking algorithm), and for converting a positive integer to its corresponding RD-sequence (i.e., unranking algorithm). Both the ranking and unranking algorithms can be run in O(tn) time and without really building any auxiliary table. ∗All correspondence should be addressed to Professor Yue–Li Wang, Department of Computer Science and Information Engineering, National Chi Nan University, University Rd. Puli, Nantou Hsien, Taiwan 545 Republic of China (Email: [email protected]).