The Variance of the height of binary search trees

By using analytic tools it is shown that the variance E(H_n−EH_n)² of the height H_n of binary search trees of size n is bounded.     

Analysis of the standard deletion algorithms in exact fit domain binary search trees

It is well known that the expected search time in anN node binary search tree generated by a random sequence of insertions isO(logN). Little has been published about the asymptotic cost when insertions and deletions are made following the usual algorithms with no attempt to retain balance. We show that after a sufficient number of updates, each consisting of choosing an element at random, removing it, and reinserting the same value, that the average search cost is (N ^1/2).This work was done in part while the first author was at the University of Waterloo. This work was supported by an NSERC '67 Science Scholarship and Grant A-8237 and the Information Technology Research Centre of Ontario.     

Efficient algorithms to create and maintain balanced and threaded binary search trees

The algorithm proposed by Chang and lyengar to perfectly balance binary search trees has been modified to not only balance but also thread binary search trees. Threads are constructed in the same sequence as normal pointers during the balancing process. No extra workspace is necessary, and the running time is also linear for the modified algorithm. Such produced tree structure has minimal average path length for fast information retrieval, and threads to facilitate more flexible and efficient traversing schemes. Maintenance and manipulation of the data structure are discussed and relevant algorithms given.     

Smoothed analysis of binary search trees

Binary search trees are one of the most fundamental data structures. While the height of such a tree may be linear in the worst case, the average height with respect to the uniform distribution is only logarithmic. The exact value is one of the best studied problems in average-case complexity.     

Online shape learning using binary search trees

In this paper we propose an online shape learning algorithm based on the self-balancing binary search tree data structure for the storage and retrieval of shape templates. This structure can also be used for classification purposes. We introduce a similarity measure with which we can make decisions on how to traverse the tree and even backtrack through the search path to find more candidate matches. Then we describe every basic operation a binary search tree can perform adapted to such a tree of shapes. Note that as a property of binary search trees, all operations can be performed in O(logn)$O(\log n)$ time and are very efficient. Finally, we present experimental data evaluating the performance of the proposed algorithm and demonstrating the suitability of this data structure for the purpose it was designed to serve.     

A method for constructing binary search trees by making insertions at the root

It is possible to construct a binary search tree by inserting items at the root instead of adding them as leaves. When used for sorting, the method has several desirable properties, including (a) fewer comparisons in the best case, (b) fewer comparisons in the worst case, (c) a reduced variance, and (d) good performance when the items are already nearly sorted or nearly reverse sorted. For applications in which the tree is searched for existing items as well as having new items added to it (e.g., in the construction of a symbol table), the tree can be made to exhibit stacklike behavior, so that the fewest comparisons are required to locate the most recently used items.     

A Markov chain model for dynamic binary search tree behaviour

We present a Markov chain model for the analysis of the behaviour of binary search trees (BSTs) under the dynamic conditions of insertions and deletions. The model is based on a data structure called a lineage tree, which provides a compact representation of different BST structures while still retaining enough information to model the effect of insertions and deletions and to compute average path length and tree height. Different lineages in the lineage tree correspond to states in the Markov chain. Transition probabilities are based on the number of BST structures corresponding to each lineage. The model is based on a similar lineage tree model developed for B-trees. The BST model is not intended for practical computations, but rather as a demonstration of the generalizability of the lineage tree approach for modeling data structures such as B-trees, B*-trees, B⁺-trees, BSTs, etc.     

A note on searching in a binary search tree

An algorithm for searching in a binary search tree using two-way comparisons is presented. The number of comparisons required by this algorithm is only one more than when using three-way comparisons. Since most high-level programming languages do not supply three-way comparisons, the number of comparisons used de facto is reduced by a factor of two. We give experimental results to indicate the speed-up that may be achieved by the presented algorithm.     

The geometry and dynamics of binary trees

The modeling of a fully populated 3D tree able to regulate dynamically remains a relatively unexplored field. A non-dimensional representation of “autoregulation” coupled with an asymmetric binary tree algorithm has been developed. The tree has a defined topology as well as a spatial representation in 3D. An analysis using a simple linearization shows the systems dynamics when perturbed away from equilibrium. Results, based on previously published work by Karch and Schreiner are presented for a variety of parameters which provide different shapes of the tree and indicate a possible mechanism for “growing” the tree in specified directions. In addition the tree, through the use of local tagging has the ability to vary its size locally via a coupled set of conservation and reverting differential equations.     

Binary search trees with binary comparison cost

We introduce a new variant of the cost measure usually associated with binary search trees. This cost measure BCOST, results from the observation that during a search, a decision to branch left need require only one binary comparison, whereas branching right or not branching at all requires two binary comparisons. This is in contrast with the standard cost measure TCOST, which assumes an equal number of comparisons is required for each of the three possible actions. With BCOST in mind we re-examine its effect with respect to minimal and maximal BCOST trees, minimal and maximal BCOST-height trees, and introduce a class of BCOST-height balanced trees, which have a logarithmically maintainable stratified subclass. Finally, a number of other issues are briefly touched upon.This work was partially supported by NATO Grant GR 155.81, Natural Sciences and Engineering Research Council of Canada Grant No. A-5692, and National Science Foundation Grant No. MCS-8116522.     

Self-organizing maps whose topologies can be learned with adaptive binary search trees using conditional rotations

Numerous variants of Self-Organizing Maps (SOMs) have been proposed in the literature, including those which also possess an underlying structure, and in some cases, this structure itself can be defined by the user. Although the concepts of growing the SOM and updating it have been studied, the whole issue of using a self-organizing Adaptive Data Structure (ADS) to further enhance the properties of the underlying SOM, has been unexplored. In an earlier work, we impose an arbitrary, user-defined, tree-like topology onto the codebooks, which consequently enforced a neighborhood phenomenon and the so-called tree-based Bubble of Activity (BoA). In this paper, we consider how the underlying tree itself can be rendered dynamic and adaptively transformed. To do this, we present methods by which a SOM with an underlying Binary Search Tree (BST) structure can be adaptively re-structured using Conditional Rotations (CONROT). These rotations on the nodes of the tree are local, can be done in constant time, and performed so as to decrease the Weighted Path Length (WPL) of the entire tree. In doing this, we introduce the pioneering concept referred to as Neural Promotion, where neurons gain prominence in the Neural Network (NN) as their significance increases. We are not aware of any research which deals with the issue of Neural Promotion. The advantage of such a scheme is that the user need not be aware of any of the topological peculiarities of the stochastic data distribution. Rather, the algorithm, referred to as the TTOSOM with Conditional Rotations (TTOCONROT), converges in such a manner that the neurons are ultimately placed in the input space so as to represent its stochastic distribution, and additionally, the neighborhood properties of the neurons suit the best BST that represents the data. These properties have been confirmed by our experimental results on a variety of data sets. We submit that all these concepts are both novel and of a pioneering sort.     

ISA [k] trees: A class of binary search trees with minimal or near minimal internal path length

In recent years several authors have investigated binary search trees with minimal internal path length. In this paper we propose relaxing the requirement of inserting all nodes on one level before going to the next level. This leads to a new class of binary search trees called ISA [k] trees. We investigated the average locate cost per node, average shift cost per node, total insertion cost, and average successful search cost for ISA[k] trees. We also present an insertion algorithm with associated predecessor and successor functions for ISA[k] trees. For large binary search trees (over 160 nodes) our results suggest the use of ISA[2] or ISA[3] trees for best performance.     

Generalized non-recursive traversal of binary trees

A non-recursive algorithm for the traversal of a binary tree is presented in which the order of traversal is defined by an external data array, allowing any of the six possible orders to be selected without modification to the algorithm itself. The extra storage requirements are three pointer variables and two bits per node (or two bits per level if an auxiliary stack is used.) The algorithm is a generalization of the pointer reversal method of Schorr and Waite and is derived by transformation of a generalized recursive version. The algorithm is described using the notation and conventions of Pascal.