and Applied Analysis 3

In this paper we consider the class of binary trees and present the results of a comprehensive experimental study on the four most representative algorithms for drawing trees, one for each of the following treedrawing approaches: Separation-Based, Path-based, Level-based, and Ringed Circular Layout. We establish a simpler, more intuitive format for storing binary trees in files and create a lar...

An algorithm of Knuth for finding an optimal binary tree is extended in several directions to solve related problems. The first case considered is restricting the depth of the tree by some predetermined integer K, and a Kn algorithm is given. Next, for trees of degree tr, rather than binary trees, Kn log tr and n log tr algorithms are found for the restricted and nonrestricted cases, respective...

We extend a classical construction on symmetric functions, the superization process, to several combinatorial Hopf algebras, and obtain analogs of the hook-content formula for the (q, t)-specializations of various bases. Exploiting the dendriform structures yields in particular (q, t)-analogs of the Björner-Wachs q-hook-length formulas for binary trees, and similar formulas for plane trees.

A binary coupling tree on n + 1 leaves is a binary tree in which the leaves have distinct labels. The rotation graph Gn is defined as the graph of all binary coupling trees on n+1 leaves, with edges connecting trees that can be transformed into each other by a single rotation. In this paper we study distance properties of the graph Gn. Exact results for the diameter of Gn for values up to n = 1...

We give a general formula for the number of occurrences of a pattern, or set of patterns, in the class of ordered (plane-planted) trees with a given number of edges. The proof is com-binatorial. Many known enumerations of ordered and binary trees are special cases of this formula.

Restricted rotation distance between pairs of rooted binary trees quantifies differences in tree shape. Cleary exhibited a linear upper bound of 12n for the restricted rotation distance between two trees with n interior nodes, and a lower bound of (n − 1)/3 if the two trees satisfy a reduction condition. We obtain a significantly improved sharp upper bound of 4n− 8 for restricted rotation dista...

The product , where and are positive integers, can be expressed as the sum of terms, each being equal to . Similarly, can be expressed as the product of factors, each being equal to . This basically works well because the sum and product operations for integers are associative; to push this process one level further (i.e. define a new operation by iterating exponentiation), one needs to decides...

Deletions in binary search trees are difficult to analyse as they are not randomness preserving. We present a new kind of tree which differs slightly from the standard binary search tree. It is referred to as an ordered binary search tree as it stores a history element in its nodes, which provides information about the order in which the nodes were inserted. Using this extra information it is p...

In this paper we present how to reconfigure dynamically a complete binary tree in faulty hypercubes. First, we use a dynamic algorithm to reconfigure a complete binary tree of height h (h≥0) in an (h+1)-dimensional faulty hypercube. If there is a faulty node in the hypercube, both the dilation and congestion are 2 after reconfiguration. If there are two faulty nodes in the hypercube, both the d...