We present a new encoding scheme for binary trees with n internal nodes whose heights are bounded by a given value h, h≧⌈log2(n +1)1⌉. The scheme encodes the internal nodes of the tree level by level and enables us to develop an algorithm for generating all binary trees within this class in a certain predetermined order. Specifically, the trees are generated in decreasing height and for trees of the same height they are generated in lexicographically increasing order. The algorithm can be easily generalized to encompass t-ary trees with bounded height. It is then shown that the average generation time per tree is constant (independent of n and h).

Generating binary trees of bounded height

In this paper we study the space of operator trees that can be used to answer a join query, with the goal of generating elements form this space at random. We solve the problem for queries with acyclic query graphs. We first count, in O(n3) time, the exact number of trees that can be used to evaluate a given query on n relations. The intermediate results of the counting procedure then serve to generate random, uniformly distributed operator trees in O(n2) time per tree. We also establish a mapping between the N operator trees for a query and the integers 1 through N —i. e. a ranking-and describe ranking and unranking procedures with complexity O(n2) and O(n2 log n), respectively.

Uniformly-distributed random generation of join orders

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Uniformly-Distributed Random Generation of Join Orders

Practical algorithms to rank necklaces, Lyndon words, and de Bruijn sequences

On generating B-trees with constant average delay and in lexicographic order

In this paper we consider the generating, ranking, and unranking of 2-3 trees with n keys. We propose a linear ordering among these trees. The problem of ranking is to determine the rank of a given tree in this ordering, while unranking means constructing the tree of a given rank. The main result is that ranking and unranking can be done in $O(n)$ time after a preprocessing step that takes $O(n^2 )$ time and space.

C. K. Wong

Papers

Ranking and Unranking of 2-3 Trees