In this paper, we propose novel representation of binary trees, named the balanced ternary-tree representation. We examine flexible division of binary trees in which we can divide a tree at any node rather than just at the root, and introduce the ternary-tree representation for the flexible division. Due to the flexibility of division, for any binary tree, balanced or ill-balanced, there is alw...

The performance of parallel computer systems is increasingly limited by constraints imposed by interconnects and this limitation will inevitably become more serious as capability of each processing node increases. Optics has already proved its worth in telecommunications and more recently in interconnecting computers and computer peripherals. In this paper, we propose an optical hybrid tree net...

What are the natural loss functions or fitting criteria for binary class probability estimation? This question has a simple answer: so-called “proper scoring rules”, that is, functions that score probability estimates in view of data in a Fisher-consistent manner. Proper scoring rules comprise most loss functions currently in use: log-loss, squared error loss, boosting loss, and as limiting cas...

Trees are a fundamental structure in algorithmics. In this paper we study the transformation of an arbitrary binary tree S with n vertices into a completely balanced tree T via rotations, a widely studied elementary tree operation. Combining concepts on rotation distance and data structures we give a basic algorithm that performs the transformation in Θ(n) time and Θ(1) space, making at most 2n...

We give linear-time algorithms for re-ordering and heightrestricting a binary search tree with only a small increase in cost, constructing a nearly optimal binary search tree given the rank by probability of each possible outcome, and height-restricting an optimal binary search tree when the increase in cost is restricted. Whereas most algorithms for constructing good binary search trees need t...

A phylogenetic tree on a label set L is a tree which exactly ILl leaves (vert.ices of degree one), no vertices of degree two, and each leaf labeled with a distinct element from L. A binary phylogenetic tree is a phylogenetic tree in which every non-leaf has degree three. Phylogenetic trees get their name because they are often appropriate models for evolutionary history. However, for simplicity...

If we insert a set of n items into a binary search tree using TREE-INSERT, the resulting tree may be horribly unbalanced. As we saw in class, however, we expect randomly built binary search trees to be balanced. (Precisely, a randomly built binary search tree has expected height O(lg n).) Therefore, if we want to build an expected balanced tree for a fixed set of items, we could randomly permut...

In this note, we give a parity reversing involution on colored binary trees which leads to a combinatorial interpretation of Formula (1.2). We make a simple variation of the bijection between colored ternary trees and binary trees proposed by Sun [2] and find a correspondence between certain class of binary trees and the set of colored 5-ary trees. The generalization of the parity reversing inv...

The problem of performing a search procedure when some of the answers to queries may be erroneous has been recently investigated by many authors (see [3,4, 2, 11). One of the main questions in this domain was stated by S. M. Ulam [S]: what is the minimal number of yes-no queries sufficient to find an unknown object in an n-element set if at most one answer may be erroneous? This problem was fir...

A chordal graph is the intersection graph of a family of subtrees of a tree, or, equivalently, it is the (edge-)intersection graph of leaf-generated subtrees of a full binary tree. In this paper, a generalization of chordal graphs from this viewpoint is studied: a graph G=(V; E) is representable if there is a family of subtrees {Sv}v∈V of a binary tree, such that uv ∈ E if and only if |Su ∩ Sv|...