CONSTANT-MEMORY ITERATIVE GENERATION OF SPECIAL STRINGS REPRESENTING BINARY TREES

Abstract

The shapes of binary trees can be encoded as permutations having a very special property. These permutations are tree permutations, or equivalently they avoid subwords of the type 231. The generation of binary trees in natural order corresponds to the generation of these special permutations in the lexicographic order. In this paper we use a stringologic approach to the generation of these special permutations: decompositions of essential parts into the subwords having staircase shapes. A given permutation differs from the next one with respect to its tail called here the working suffix. Some new properties of such working suffixes are discovered in the paper and used to design effective algorithms transforming one tree permutation into its successor or predecessor in the lexicographic order. The algorithms use a constant amount of additional memory and they look only at those elements of the permutation which belong to the working suffix. The best-case, average-case and worst-case time complexities of the algorithms are O(1), O(1), and O(n) respectively. The advantages of our stringologic approach are constant time and iterative generation, while other known algorithms are usually recursive or not constant-memory ones. In this paper we also present a new compact non-recursive linear time algorithm solving a related problem of decoding the shape of a binary tree from its corresponding tree permutation.