| Summary: | 碩士 === 國立臺灣科技大學 === 資訊管理系 === 95 === The enumeration of binary trees and t-ary trees has been extensively discussed in recent literature. Binary trees and t-ary trees are called regular trees. A lot of researchers have introduced integer sequences or permutations to characterize regular trees and provided efficient algorithms to generate those sequences in lexicographic order.
In this thesis, we are concerned with the enumeration of uncertain-ary trees in lexicographic order. A tree is said to be uncertain-ary if two nodes in different levels may have a different number of children. Then, we use a concise representation, called right distance sequences (or RD-sequence for short), to describe all uncertain-ary trees with n internal nodes. Using an uncertain-ary recursion tree and its concomitant tables, a systematical way can help us to investigate the structural representations of uncertain-ary trees. In addition, an algorithm is developed to enumerate all uncertain-ary trees with n internal nodes using RD-sequences. We also present two algorithms for determining the rank of a given uncertain-ary tree in lexicographic order (i.e., a ranking algorithm), and for converting a positive integer to its corresponding RD-sequence (i.e., an unranking algorithm).
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