By embedding multiple proximal SVM classifiers into a binary tree architecture, it is possible to turn an arbitrary multi-classes problem into a hierarchy of binary classifications. The critical issue then consists in determining in each node of the tree how to aggregate the multiple classes into a pair of say overlay classes to discriminate. As a fundamental contribution, our paper proposes to...

Given a sequence of numbers ul,. ,a,,, find a binary tree with q leaves minimizing max{/l, + (II,. , II,, + u, }, where h, is the distance from the ith leaf to the root, i = I, , q. This problem is solved by means of a O(q) algorithm and a tight upper bound for the minimum is given by an explicit formula. The task is equivalent to finding a binary tree of minimum height having y subtrees of hei...

One can prove the following propositions: (1) For every set D and for every finite sequence p and for every natural number n such that p ∈ D holds p↾ Segn ∈ D. (2) For every binary tree T holds every element of T is a finite sequence of elements of Boolean. Let T be a binary tree. We see that the element of T is a finite sequence of elements of Boolean. Next we state several propositions: (3) F...

Three standard subtree transfer operations for binary trees, used in particular for phylogenetic trees, are: tree bisection and reconnection (TBR), subtree prune and regraft (SPR) and rooted subtree prune and regraft (rSPR). For a pair of leaf-labelled binary trees with n leaves, the maximum number of such moves required to transform one into the other is n−Θ( √ n), extending a result of Ding, ...

We investigate some attractive features of the 1-D sequence of pixels produced by the peano traversal of an image. We introduce two new hardware operations called bit-spreaded-meshing and its inverse bit-collation to produce and invert the sequence in real-time. A compact binary tree built using this sequence at its base implicitly contains the well known quadtree of the image also. The binary ...

In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled trees, and labeled plane trees.

In this paper we show that, when a binary tree is in a certain critical balance, there emerge the Golden Ratio and the Fibonacci numbers. The paper consists of two sections. In the first section we find some elementary balance properties of optimal binary trees with variously weighted leaves. In the second section, a basic inequality implied by the optimality of trees is in turn used to define ...

The binary search tree or Quicksort is the most used of all sorting algorithm, since it is both fast and simple. The random recursive tree is another extensively studied random tree. We have examined fringe trees in these two types of random trees, since the study of such subtrees appears to be an effective approach towards defining characteristics also of the whole tree. By using a representat...

The number b, of binary trees with n nodes is well known. Two ways to determine b, are presented in [3]. One, based on constructing them, leads to recurrence relations solved by generating functions. The other recognizes that b, is the same as the number of stack permutations and counts these permutations. In [l], a method based on well-formed sequences of parentheses is given, and the result i...

A binary tree is a common data structure used in many branches of computer science. It is often used for solving various types of searching problems. In complexity analysis we usually abstract from the real implementation and derive easily the time complexity of traversal from the root of a balanced tree to any leaf in asymptotic time O(log(m)), where m is the number of leaves. In this paper we...