The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance

DOI10.1214/105051606000000547zbMath1124.05025arXivmath/0702415OpenAlexW3100743224MaRDI QIDQ997423

Svante Janson, Michael G. B. Blum, Olivier Francois

Publication date: 6 August 2007

Published in: The Annals of Applied Probability (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0702415

zbMATH Keywords

central limit theoremcontraction methodYule processshape statisticsAiry distributionCatalan treesrandom phylogenetic trees

Mathematics Subject Classification ID

Applications of statistics to biology and medical sciences; meta analysis (62P10) Central limit and other weak theorems (60F05) Trees (05C05) Problems related to evolution (92D15) Combinatorial probability (60C05)

Related Items (26)

Expected Number of Induced Subtrees Shared by Two Independent Copies of a Random Tree ⋮ Stochastic analysis of the extra clustering model for animal grouping ⋮ The fixed points of the multivariate smoothing transform ⋮ Squaring within the Colless index yields a better balance index ⋮ On finding the Adams consensus tree ⋮ Distributional analysis of the extra-clustering model with uniformly generated phylogenetic trees ⋮ On the minimum value of the Colless index and the bifurcating trees that achieve it ⋮ Hardness of \textsc{Balanced Mobiles} ⋮ A Mathematical Connection Between Single-Elimination Sports Tournaments and Evolutionary Trees ⋮ Exact formulas for the variance of several balance indices under the Yule model ⋮ Limit distribution of the quartet balance index for Aldous’s $(\beta \ge 0)$-model ⋮ Exact and approximate limit behaviour of the Yule tree's cophenetic index ⋮ Heavy tailed solutions of multivariate smoothing transforms ⋮ Gaussian approximations for phylogenetic branch length statistics under stochastic models of biodiversity ⋮ On a matching distance between rooted phylogenetic trees ⋮ Cost functionals for large (uniform and simply generated) random trees ⋮ Revisiting Shao and Sokal's \(B_2\) index of phylogenetic balance ⋮ Maximal clades in random binary search trees ⋮ Probability distributions of ancestries and genealogical distances on stochastically generated rooted binary trees ⋮ Metric dimension of critical Galton-Watson trees and linear preferential attachment trees ⋮ Large deviations for the leaves in some random trees ⋮ A simple derivation of the mean of the Sackin index of tree balance under the uniform model on rooted binary labeled trees ⋮ Limit theorems for patterns in phylogenetic trees ⋮ Two results about the Sackin and Colless indices for phylogenetic trees and their shapes ⋮ Clades and clans: a comparison study of two evolutionary models ⋮ The distributions under two species-tree models of the number of root ancestral configurations for matching gene trees and species trees

Uses Software

• Quicksort

Cites Work

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• Limiting distributions for additive functionals on Catalan trees
• Asymptotic fringe distributions for general families of random trees
• The coalescent
• On the number of descendants and ascendants in random search trees
• Stochastic models and descriptive statistics for phylogenetic trees, from Yule to today.
• On statistical tests of phylogenetic tree imbalance: The Sackin and other indices revisited
• On a multivariate contraction method for random recursive structures with applications to Quicksort
• Phase Change of Limit Laws in the Quicksort Recurrence under Varying Toll Functions
• A bernoulli excursion and its various applications
• The Wiener Index of simply generated random trees
• The evolution of a plane mixing layer with spanwise nonuniform forcing
• Probability metrics and recursive algorithms
• The probabilities of rooted tree-shapes generated by random bifurcation
• A limit theorem for “quicksort”
• Quicksort
• Properties of phylogenetic trees generated by Yule-type speciation models
• Analytic variations on the Airy distribution

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