A Note on Some Tree Similarity Measures

Information Processing Letters, 1983

Olivii has recently introduced the class of 'half-balanced' binary search trees, which have O(log n) access time but require only a constant number of single rotations for rebalancing after an insertion or a deletion. In this paper we show that a well-known class of balanced binary trees, the 'symmetric binary B-trees' of Bayer, have the same properties. This is not surprising, for Bayer's class and Olivit-'s class contain exactly the same binary trees.

ARTICLE INFO Binary trees are essential structures in Computer Science. The leaf (leaves) of a binary tree is one of the most significant aspects of it. In this study, we prove that the order of a leaf (leaves) of a binary tree is the same in the main tree traversals; preorder, inorder, and postorder. Then, we prove that given the preorder and postorder traversals of a binary tree, the leaf (leaves) of a binary tree can be determined. We present the algorithm BT-leaf, a novel one, to detect the leaf (leaves) of a binary tree from its preorder and postorder traversals in quadratic time and linear space.

Journal of Algorithms and Computation, 2017

Binary trees are essential structures in Computer Science. The leaf (leaves) of a binary tree is one of the most significant aspects of it. In this study, we prove that the order of a leaf (leaves) of a binary tree is the same in the main tree traversals; preorder, inorder, and postorder. Then, we prove that given the preorder and postorder traversals of a binary tree, the leaf (leaves) of a binary tree can be determined. We present the algorithm BT-leaf, a novel one, to detect the leaf (leaves) of a binary tree from its preorder and postorder traversals in quadratic time and linear space. http://jac.ut.ac.ir/article_406_49.html

1997

This article introduces the basic concepts of binary trees, and then works through a series of practice problems with solution code in C/C++ and Java. Binary trees have an elegant recursive pointer structure, so they are a good way to learn recursive pointer algorithms.

IEEE Transactions on Knowledge and Data Engineering, 1993

Consider a set A = {A 1 , A 2 ,...,A N } of records, where each record is identified by a unique key. The records are accessed based on a set of access probabilities S = [s 1 ,s 2 ,...,s N ] and are to be arranged lexicographically using a Binary Search Tree (BST). If S is known a priori, it is well known [10] that an optimal BST may be constructed using A and S. We consider the case when S is not known a priori . A new restructuring heuristic is introduced that requires three extra integer memory locations per record. In this scheme the restructuring is performed only if it decreases the Weighted Path Length (WPL) of the overall resultant tree. An optimized version of the latter method which requires only one extra integer field per record has also been presented.

Acta Cybernetica, 2019

Let T be a rooted directed tree on n vertices, rooted at v. The rooted subtree frequency vector (RSTF-vector) of T with root v, denoted by rstf(T, v) is a vector of length n whose entry at position k is the number of subtrees of T that contain v and have exactly k vertices. In this paper we present an algorithm for reconstructing rooted directed trees from their rooted subtree frequencies (up to isomorphism). We show that there are examples of nonisomorphic pairs of rooted directed trees that are RSTF-equivalent, that is they share the same rooted subtree frequency vectors. We have found all such pairs (groups) for small sizes by using exhaustive computer search. We show that infinitely many nonisomorphic RSTF-equivalent pairs of trees exist by constructing infinite families of examples.

The paper introduces an operation on a binary tree, called binary tree roll, or roll of a binary tree. Two versions of the binary tree roll, counterclockwise and clockwise, are presented. The operations are mathematically defined and graphically presented. It is explained how the binary tree roll actually coincides with the process of turning the entire tree 90 degrees counterclockwise or clockwise. To visually explain and perform the roll operation, the concepts of a wedge node, true ancestor, illusory ancestor, illusory root and illusory ancestral stem of nodes are introduced, as well as the visual operations of turning and downshift. Both roll operations are implemented using programming algorithms. The algorithms are explained, and all the situations that might be encountered during processing the roll operation are examined and resolved. Thus, the paper gives a mathematical introduction of both binary tree roll operations, gives their visual explanations and offers algorithms for their implementations using a computer.

Theoretical Computer Science, 2007

Trees are probably the most studied class of graphs in Computer Science. In this thesis we study bijective codes that represent labeled trees by means of string of node labels. We contribute to the understanding of their algorithmic tractability, their properties, and their applications.

Information Processing Letters, 1977

Information Processing Letters, 2010

The rotation distance d(S, T) between two binary trees S, T of n vertices is the minimum number of rotations to transform S into T. While it is known that d(S, T) 2n − 6, a wellknown conjecture states that there are trees for which this bound is sharp for any value of n 11. We are unable to prove the conjecture, but we give here some simple criteria for lower bound evaluation, leading for example to individuate some "regular" tree structures for which d(S, T) = 3n/2 − O (1), or d(S, T) = 5n/3 − O (1).

In this paper we propose a dynamic programming algorithm to evaluate local similarity between unordered trees using a constrained edit distance. The core of the method relies on an adaptation of an algorithm proposed by Zhang for comparing unordered rooted trees. The method is currently being used in plant architecture modeling to outline similar sub-parts on plant structure and in genomic analysis to evaluate variability between secondary structures of RNA.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2014

Theoretical Computer Science, 1999

Inferring evolutionary trees is an interesting and important problem in biology, but one that is computationally difficult as most associated optimization problems are NP-hard. Although many methods are provably statistically consistent (i.e. the probability of recovering the correct tree converges to 1 as the sequence length increases), the actual rate of convergence for different methods has not been well understood. In a recent paper we introduced a new method for reconstructing evolutionary trees called the dyadic closure method (DCM), and we showed that DCM has a very fast convergence rate. DCM runs in O(n 5 logn) time, where n is the number of sequences, and so, although polynomial, the computational requirements are potentially too large to be of use in practice. In this paper we present another tree reconstruction method, the witness-antiwitness method (WAM). WAM is faster than DCM, especially on random trees, and converges to the true tree topology at the same rate as DCM. We also compare WAM to other methods used to reconstruct trees, including Neighbor Joining (possibly the most popular method among molecular biologists), and new methods introduced in the computer science literature.