Consider the complete rooted binary tree T. We construct a sequence Tn, n = 1, 2, . . . of subtrees of T recursively as follows. T1 consists only of the root. Given Tn, we choose a leaf u uniformly at random from the set of all leaves of Tn and add its two children to the tree to create Tn+1. Thus Tn+1 consists of Tn and the children u1, u2 of u, and contains in total 2n+ 1 nodes, including n+ ...

This paper empirically compares five linear-time algorithms for generating unbiased random binary trees. More specifically, we compare the relative asymptotic performance of the algorithms in terms of the numbers of various basic operations executed on average per tree node. Based on these numbers a ranking of the algorithms, which depends on the operation weights, can be deduced. A definitive ...

We study tree series and weighted tree automata over unranked trees. The message is that recognizable tree series for unranked trees can be defined and studied from recognizable tree series for binary representations of unranked trees. For this we prove results of [1] as follows. We extend hedge automata – a class of tree automata for unranked trees – to weighted hedge automata. We define weigh...

It has been shown that the cost W of a weight balanced binary tree satisfies the inequalities, H 5 W 5 H + 3, where H is the entropy of the set of the leaves. For a class of “smooth” distributions the inequalities, H 5 W 5 H + 2, are derived. These results imply that for sets with large entropy the search times provided by such trees cannot be substantially shortened when binary decisions are b...

We study the number of records in a random binary search tree on n randomly labelled vertices. Equivalently the number of random cuttings required to eliminate a random binary search tree can be studied. After normalization the distribution is shown to be asymptotically 1-stable.

In this paper we derive a linear-time, constant-space algorithm to construct a binary heap whose inorder traversal equals a given sequence. We do so in two steps. First, we invert a program that computes the inorder traversal of a binary heap, using the proof rules for program inversion by W. Chen and J.T. Udding. This results in a linear-time solution in terms of binary trees. Subsequently, we...

We present a performance comparison of tree data structures for N -body simulation. The tree data structures examined are the balanced binary tree and the Barnes– Hut (BH) tree. Previous work has compared the performance of BH trees with that of nearest-neighbor trees and the fast multipole method, but the relative merits of BH and binary trees have not been compared systematically. In carrying...

Binary Space Partition (BSP) tree and Constructive Solid Geometry (CSG) tree representations are both set-theoretic binary tree representations of solid objects used in solid modeling and computer graphics. Recently, an extension of the traditional BSP tree definition has been presented, in which surfaces used in the binary partition include curved surfaces in addition to planar surfaces. We ex...

Preferences can be aggregated using voting rules. We consider here the family of rules which perform a sequence of pairwise majority comparisons between two candidates. The winner thus depends on the chosen sequence of comparisons, which can be represented by a binary tree. We address the difficulty of computing candidates that win for some trees, and then introduce and study the notion of fair...

Note that k must always equal i+ 1 in a binary tree. Prodinger [P] recently computed the probability that a random binary tree with n nodes has i nodes with 2 children (and hence i + 1 nodes without children and n − 2i − 1 nodes with 1 child). Since the total number of binary trees with n nodes is known—it is bn—, his formulas can be derived easily from the above theorem and vice versa. Proding...