scholarly journals On the distribution of winning moves in random game trees

1981 ◽  
Vol 24 (2) ◽  
pp. 227-237
Author(s):  
J. A. Flanigan
Keyword(s):  
Asymptotic Distribution ◽  
Rooted Tree ◽  
Random Variable ◽  
Initial Position ◽  
Rooted Trees ◽  
Contour Integral ◽  
Game Trees ◽  
Vertex Set ◽  
Natural Way

In a natural way, associated with each rooted tree there exists a pair of two-person games, each game possessing the root as initial position. When a rooted tree is selected at random from the set of all rooted trees which possess {1, 2, …, n} as vertex set, the number of winning moves available to the first player to move in each of the associated games is a random variable. For fixed n, we determine the distribution of this random variable. As an immediate consequence, we find the probability that the first player to move has no winning move at all. The “saddlepoint method” is applied to a certain contour integral to obtain the asymptotic distribution of the number of winning moves as n → ∞.

scholarly journals Counting Forests by Descents and Leaves

10.37236/1266 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Ira M. Gessel
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Exponential Generating Function ◽  
Vertex Set

A descent of a rooted tree with totally ordered vertices is a vertex that is greater than at least one of its children. A leaf is a vertex with no children. We show that the number of forests of rooted trees on a given vertex set with $i+1$ leaves and $j$ descents is equal to the number with $j+1$ leaves and $i$ descents. We do this by finding a functional equation for the corresponding exponential generating function that shows that it is symmetric.

Random flights on regular graphs

1984 ◽  
Vol 16 (03) ◽  
pp. 618-637 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Automorphism Group ◽  
Transition Probabilities ◽  
Finite Graph ◽  
Initial Position ◽  
Regular Graphs ◽  
Random Flights ◽  
The Past ◽  
Initial Location ◽  
Vertex Set

Let K be a finite graph with vertex set V = {x 0, x 1, …, xσ –1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by vn (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v 0 the initial location. It is assumed that the transition probabilities P{vn = xj | vn –1 = xi }, xi ϵ V, xj ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.

The asymptotic distribution of random molecules

1980 ◽  
Vol 12 (03) ◽  
pp. 640-654
Author(s):  
Wulf Rehder
Keyword(s):  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Random Variable ◽  
Unit Cube ◽  
Image Position ◽  
Solid Spheres

If n solid spheres K n of some volume V(K n ) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form L n (s) molecules with exactly s atoms K n. The random variable L n(s) has a limit distribution if V(K n ) tends to zero but nV(Kn ) tends to infinity at a certain rate: it is shown that for L n(s) is asymptotically Poisson.

The asymptotic distribution of random molecules

1980 ◽  
Vol 12 (3) ◽  
pp. 640-654 ◽  
Author(s):  
Wulf Rehder
Keyword(s):  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Random Variable ◽  
Unit Cube ◽  
Image Position ◽  
Solid Spheres

If n solid spheres Kn of some volume V(Kn) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form Ln(s) molecules with exactly s atoms Kn. The random variable Ln(s) has a limit distribution if V(Kn) tends to zero but nV(Kn) tends to infinity at a certain rate: it is shown that for Ln(s) is asymptotically Poisson.

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.

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.

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