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Binary Tree Sequence Rotations and t-ary Tree Enumerations
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In this book, we consider a transformation on binary§trees using new types of rotations. Each of the newly§proposed rotations is permitted only at nodes on the§left-arm or the right-arm of a tree. Consequently, we§develop a linear time algorithm with at most n 1§rotations for converting weight sequences between any§two binary trees.§we use right distance sequences (or RD-sequences for§short), to describe all t-ary trees with n internal§nodes. Using a t-ary 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 in§lexicographic order (i.e., the ranking algorithm),§and for converting a positive integer to its§corresponding RD-sequence (i.e., the unranking§algorithm). Both the ranking and unranking algorithms§can be run in O(tn) time and without really building§any auxiliary table. In addition, we also present a§loopless algorithm to enumerate Gray-codes of t-ary§trees using RD-sequences.
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