One of the criteria for inferring a species tree from a collection of gene trees, when gene tree incongruence is assumed to be due to incomplete lineage sorting (ILS), is minimize deep coalescence, or MDC. Exact algorithms for inferring the species tree from rooted, binary trees under MDC were recently introduced. Nevertheless, in phylogenetic analyses of biological data sets, estimated gene tr...

Wilke’s tree algebra formalism for characterizing families of tree languages is based on six operations involving letters, binary trees and binary contexts. In this paper a completeness property of these operations is studied. It is claimed that all functions involving letters, binary trees and binary contexts which preserve all syntactic tree algebra congruence relations of tree languages are ...

We define a D2CS of a graph G to be a set S ⊆ V (G) with diam(G[S]) ≤ 2. A D2CS arises in connection with conditional coloring and radio-k-coloring of graphs. We study the problem of counting and enumerating D2CS of a graph. We first prove the following propositions: (1) Let f (k, h) be the number of D2CS of a complete k-ary tree of height h. Then f (k, h) = k k−1 (f (k + 1, 1) − 4)(k h−1 − 1) ...

We consider the use of a static context tree for binary image compression. The contexts are stored in the leaves of a variable-depth binary tree. The tree structure itself is fully static and optimized off-line for a training image. The structure of the tree is similar for different images of the same type. The benefit from optimizing the tree for each input image separately is usually overweig...

Reconciliation is a method widely used to infer the evolutionary relationship between the members of a gene family. It consists of comparing a gene tree with a species tree, and interpreting the incongruence between the two trees as evidence of duplication and loss. In the case of binary rooted trees, linear-time algorithms have been developed for the duplication, loss, and mutation (duplicatio...

Fractals and continued fractions seem to be deeply related in many ways. Farey fractions appear naturally in both. Much of this relationship can be explained through the fact that a certain subset of the general linear group GL(2,Z) over the integers; called the “dyadic monoid”, “dyadic groupoid”, or “dyadic lattice”, is the natural symmetry of many fractals, including those associated with per...

A rotation in a binary tree is a local restructuring that changes the tree into another and preserves the inorder sequence. The rotation distance between two binary trees is the minimum number of rotations needed to transform one into another. However, a polynomial-time algorithm for computing rotation distances between any two binary trees has still not been found. Lucas (The Computer Journal,...

This paper discusses the implementation of the binary coding scheme and its comparison with the post-order, preorder and in-order traversal techniques to test for ancestor-descendant relationships in k-ary trees (a tree in which any leaf node has up to k children). The approach used is assigning a unique binary code to each node in a tree. The value of the binary code for a node is the concaten...

Many reconstruction algorithms for binary tree have been discussed in this paper. A particular focus of this paper is on “A new Non-Recursive Algorithm for Reconstructing a Binary Tree from its Traversals”. The computation time required for executing the reconstruction algorithm are O(N) and space complexity is O(NlogN) where N is the number of nodes in the binary tree. This algorithm works wel...

This paper introduces the notion of pattern avoidance in binary trees. We provide an algorithm for computing the generating function that counts the number of n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting mechanism, we study the analogue of Wilf equivalence in which two tree patterns are equivalent if the respective trees that avoid them are equinumerou...