A widely used method for constructing phylogenetic trees is the neighbour-joining method of Saitou and Nei. We develope heuristics for speeding up the neighbour-joining method which generate the same phylogenetic trees as the original method. All heuristics are based on using a quad-tree to guide the search for the next pair of nodes to join, but differ in the information stored in quad-tree no...

This paper reviews P´olya urn models and their connection to random trees. Basic results are presented, together with proofs that underly the historical evolution of the accompanying thought process. Extensions and generalizations are given according to chronology: • P´olya-Eggenberger’s urn • Bernard Friedman’s urn • Generalized P´olya urns • Extended urn schemes • Invertible urn schemes ...

This paper introduces entropy quad-trees, which are structures derived from quad-trees by allowing nodes to split only when those correspond to sufficiently complex sub-domains of a data domain. Complexity is evaluated using an information-theoretic measure based on the analysis of the entropy associated to sets of objects designated by nodes. An alternative measure related to the concept of bo...

In this paper, a new approach (scheme) to the analysis of quad-trees in the discrete wavelet spectrum of a digital image is proposed. During the pre-scanning phase, the proposed scheme generates problem-oriented binary codes for the whole set of quad-tree roots (wavelet coefficients) and thereby accumulates information on the significance of respective descendants (wavelet coefficients comprisi...

Approaches to polygonal decomposition for hierarchical image representation are described. For planar decomposition, quad trees using square and triangular neighborhoods are found to be the only feasible methods, having the same computational complexity. For grid images the choice of the appropriate tree type is determined by the grid topology. Triangular and square quad trees are appropriate f...