The probabilities of rooted tree-shapes generated by random bifurcation

1971 ◽  
Vol 3 (1) ◽  
pp. 44-77 ◽  
Author(s):  
E. F. Harding
Keyword(s):  
Probability Distribution ◽  
Statistical Approach ◽  
Rooted Tree ◽  
Difficult Problem ◽  
Numerical Results ◽  
Rooted Trees ◽  
Evolutionary Trees ◽  
Mathematical Problems ◽  
Labelled Trees ◽  
Random Bifurcation

The set of rooted trees, generated by random bifurcation at the terminal nodes, is considered with the aims of enumerating it and of determining its probability distribution. The account of enumeration collates much previous work and attempts a complete perspective of the problems and their solutions. Asymptotic and numerical results are given, and some unsolved problems are pointed out. The problem of ascertaining the probability distribution is solved by obtaining its governing recurrence equation, and numerical results are given. The difficult problem of determining the most probable tree-shape of given size is considered, and for labelled trees a conjecture at its solution is offered. For unlabelled shapes the problem remains open. These mathematical problems arise in attempting to reconstruct evolutionary trees by the statistical approach of Cavalli-Sforza and Edwards.

The probabilities of rooted tree-shapes generated by random bifurcation

1971 ◽  
Vol 3 (01) ◽  
pp. 44-77 ◽  
Author(s):  
E. F. Harding
Keyword(s):  
Probability Distribution ◽  
Statistical Approach ◽  
Rooted Tree ◽  
Difficult Problem ◽  
Numerical Results ◽  
Rooted Trees ◽  
Evolutionary Trees ◽  
Mathematical Problems ◽  
Labelled Trees ◽  
Random Bifurcation

The set of rooted trees, generated by random bifurcation at the terminal nodes, is considered with the aims of enumerating it and of determining its probability distribution. The account of enumeration collates much previous work and attempts a complete perspective of the problems and their solutions. Asymptotic and numerical results are given, and some unsolved problems are pointed out. The problem of ascertaining the probability distribution is solved by obtaining its governing recurrence equation, and numerical results are given. The difficult problem of determining the most probable tree-shape of given size is considered, and for labelled trees a conjecture at its solution is offered. For unlabelled shapes the problem remains open. These mathematical problems arise in attempting to reconstruct evolutionary trees by the statistical approach of Cavalli-Sforza and Edwards.

scholarly journals The distribution of wasted spaces in the M/M/∞ queue with ranked servers

2008 ◽  
Vol 40 (03) ◽  
pp. 835-855 ◽  
Author(s):  
Eunju Sohn ◽  
Charles Knessl
Keyword(s):  
Probability Distribution ◽  
Numerical Results ◽  
Tail Behavior ◽  
Service Rates ◽  
The Difference

We consider the M/M/∞ queue with m primary servers and infinitely many secondary servers. All the servers are numbered and ordered. An arriving customer takes the lowest available server. We define the wasted spaces as the difference between the highest numbered occupied server and the total number of occupied servers. Letting ρ = λ0/μ be the ratio of arrival to service rates, we study the probability distribution of the wasted spaces asymptotically for ρ → ∞. We also give some numerical results and the tail behavior for ρ = O(1).

scholarly journals The statistical physics of iceberg calving and the emergence of universal calving laws

2011 ◽  
Vol 57 (201) ◽  
pp. 3-16 ◽  
Author(s):  
J.N. Bassis
Keyword(s):  
Probability Distribution ◽  
Statistical Physics ◽  
Difficult Problem ◽  
Large Spatial Scale ◽  
Ice Shelves ◽  
Environmental Regimes ◽  
Discrete Fracture ◽  
Iceberg Calving ◽  
Semi Empirical ◽  
Scale Limit

AbstractDetermining a calving law valid for all glaciological and environmental regimes has proven to be a difficult problem in glaciology. For this reason, most models of the calving process are semi-empirical, with little connection to the underlying fracture processes. In this study, I introduce methods rooted in statistical physics to show how calving laws, valid for any glaciological domain, can emerge naturally as a large-spatial-scale/long-temporal-scale limit of an underlying continuous or discrete fracture process. An important element of the method developed here is that iceberg calving is treated as a stochastic process and that the probability an iceberg will detach in a given interval of time can be described by a probability distribution function. Using limiting assumptions about the underlying probability distribution, the theory is shown to be able to simulate a range of calving styles, including the sporadic detachment of large, tabular icebergs from ice tongues and ice shelves and the more steady detachment of smaller-sized bergs from tidewater/outlet glaciers. The method developed has the potential to provide a physical basis to include iceberg calving into numerical ice-sheet models that can be used to produce more realistic estimates of the glaciological contribution to sea-level rise.

scholarly journals On Sackin's original proposal: The variance of the leaves' depths as a phylogenetic balance index

2019 ◽  
Author(s):  
Tomás Martínez Coronado ◽  
Arnau Mir ◽  
Francesc Rossello ◽  
Lucía Rotger
Keyword(s):  
Phylogenetic Tree ◽  
Phylogenetic Trees ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Closed Formula ◽  
Original Proposal ◽  
Yule Model ◽  
Uniform Model ◽  
Rooted Phylogenetic Tree ◽  
Balance Index

Abstract Background: The Sackin index S of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's 1972 paper is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their "variation". This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin, where moreover they posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin's original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal.Results: In this paper we study the properties of the variance of the leaves' depths, V, as a balance index. Firstly, we prove that the rooted trees with n leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space BT_n of bifurcating rooted phylogenetic trees with n< 184 leaves at the so-called "maximally balanced trees" with n leaves, this property fails for almost every n>= 184. We provide then an algorithm that finds in O(n) time the trees in BT_n with minimum V value. Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance of the Sackin index and the total cophenetic indexof a bifurcating rooted tree, as well as of their covariance, under the uniform model, thus filling this gap in the literature.Conclusions: The phylogenetics crowd has been wise in preferring as a balance index the sum S(T) of the leaves’ depths of a phylogenetic tree T over their variance V (T), because the latter does not seem to capture correctly the notion of balance of large bifurcating rooted trees. But for bifurcating trees up to 183 leaves, V is a valid and useful balance index.

scholarly journals A Refinement of Cayley's Formula for Trees

10.37236/1884 ◽  
2006 ◽  
Vol 11 (2) ◽  
Author(s):  
Ira M. Gessel ◽  
Seunghyun Seo
Keyword(s):  
Rooted Tree ◽  
Rooted Trees ◽  
Binary Trees ◽  
Parking Functions ◽  
Hook Length ◽  
Proper Vertex

A proper vertex of a rooted tree with totally ordered vertices is a vertex that is the smallest of all its descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials $$P_n(a,b,c)= c\prod_{i=1}^{n-1}(ia+(n-i)b +c),$$ which reduce to $(n+1)^{n-1}$ for $a=b=c=1$. Our study of proper vertices was motivated by Postnikov's hook length formula $$(n+1)^{n-1}={n!\over 2^n}\sum _T \prod_{v}\left(1+{1\over h(v)}\right),$$ where the sum is over all unlabeled binary trees $T$ on $n$ vertices, the product is over all vertices $v$ of $T$, and $h(v)$ is the number of descendants of $v$ (including $v$). Our results give analogues of Postnikov's formula for other types of trees, and we also find an interpretation of the polynomials $P_n(a,b,c)$ in terms of parking functions.

Realization of plucking polynomials

2017 ◽  
Vol 26 (02) ◽  
pp. 1740016 ◽  
Author(s):  
Zhiyun Cheng ◽  
Sujoy Mukherjee ◽  
Józef H. Przytycki ◽  
Xiao Wang ◽  
Seung Yeop Yang
Keyword(s):  
Rooted Tree ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rooted Trees ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for a given polynomial to be a plucking polynomial of a rooted tree. We discuss the fact that different rooted trees can have the same polynomial.

scholarly journals A Beta-splitting model for evolutionary trees

2016 ◽  
Vol 3 (5) ◽  
pp. 160016 ◽  
Author(s):  
Raazesh Sainudiin ◽  
Amandine Véber
Keyword(s):  
Continuous Time ◽  
Rooted Tree ◽  
Tree Topology ◽  
Sister Species ◽  
Evolutionary Trees ◽  
Diversification Rates ◽  
Branch Lengths ◽  
The Creation ◽  
Natural Way

In this article, we construct a generalization of the Blum–François Beta-splitting model for evolutionary trees, which was itself inspired by Aldous' Beta-splitting model on cladograms. The novelty of our approach allows for asymmetric shares of diversification rates (or diversification ‘potential’) between two sister species in an evolutionarily interpretable manner, as well as the addition of extinction to the model in a natural way. We describe the incremental evolutionary construction of a tree with n leaves by splitting or freezing extant lineages through the generating, organizing and deleting processes. We then give the probability of any (binary rooted) tree under this model with no extinction, at several resolutions: ranked planar trees giving asymmetric roles to the first and second offspring species of a given species and keeping track of the order of the speciation events occurring during the creation of the tree, unranked planar trees , ranked non-planar trees and finally ( unranked non-planar ) trees . We also describe a continuous-time equivalent of the generating, organizing and deleting processes where tree topology and branch lengths are jointly modelled and provide code in SageMath/Python for these algorithms.

scholarly journals The enumeration of rooted trees by total height

1969 ◽  
Vol 10 (3-4) ◽  
pp. 278-282 ◽  
Author(s):  
John Riordan ◽  
N. J. A. Sloane
Keyword(s):  
Neural Networks ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Total Height ◽  
Random Neural Networks

The height (as in [3] and [4]) of a point in a rooted tree is the length of the path (that is, the number of lines in the path) from it to the root; the total height of a rooted tree is the sum of the heights of its points. The latter arises naturally in studies of random neural networks made by one of us (N.J.A.S.), where the enumeration of greatest interest is that of trees with all points distinctly labeled.

scholarly journals On an Identity for the Cycle Indices of Rooted Tree Automorphism Groups

10.37236/1152 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Stephan G. Wagner
Keyword(s):  
Symmetric Group ◽  
Elementary Proof ◽  
Automorphism Groups ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Cycle Index ◽  
Spanning Forest ◽  
Cycle Indices

This note deals with a formula due to G. Labelle for the summed cycle indices of all rooted trees, which resembles the well-known formula for the cycle index of the symmetric group in some way. An elementary proof is provided as well as some immediate corollaries and applications, in particular a new application to the enumeration of $k$-decomposable trees. A tree is called $k$-decomposable in this context if it has a spanning forest whose components are all of size $k$.

scholarly journals On Sackin's original proposal: The variance of the leaves' depths as a phylogenetic balance index

2020 ◽  
Author(s):  
Tomás Martínez Coronado ◽  
Arnau Mir ◽  
Francesc Rossello ◽  
Lucía Rotger
Keyword(s):  
Phylogenetic Tree ◽  
Phylogenetic Trees ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Closed Formula ◽  
Original Proposal ◽  
Yule Model ◽  
Uniform Model ◽  
Rooted Phylogenetic Tree ◽  
Balance Index

Abstract Background. The Sackin index S of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's 1972 paper is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their ``variation''. This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin in 1993, where they also posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin's original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal. Results. In this paper we study the properties of the variance of the leaves' depths, V, as a balance index. Firstly, we prove that the rooted trees with $n$ leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space of bifurcating rooted phylogenetic trees with at most 183 leaves at the so-called ``maximally balanced trees'' with n leaves, this property fails for almost every n larger than 184 We provide then an algorithm that finds the bifurcating rooted trees with n leaves and minimum V value in quasilinear time. Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance under the uniform model of the Sackin index and the total cophenetic index of a bifurcating rooted tree, as well as of their covariance, thus filling this gap in the literature.

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