On Major and Minor Branches of Rooted Trees

1987 ◽  
Vol 39 (3) ◽  
pp. 673-693 ◽  
Author(s):  
A. Meir ◽  
J. W. Moon
Keyword(s):  
Rooted Tree ◽  
Rooted Trees ◽  
Primary Branch ◽  
Secondary Branch ◽  
Not Given ◽  
Tertiary Branch ◽  
Image Position

Let denote a rooted tree with n nodes. (For definitions not given here, see, e.g. [4]). For any node v of , let B(v) denote the subtree of determined by v and all nodes u such that v is between u and the root of ; node v serves as the root of B(v). The branches of are the subtrees B(v) such that node v is joined to the root of . A branch B with i nodes is a primary branch of if n/2 ≦ i ≦ n – 1; if has a primary branch B with i nodes, then a branch C with j nodes is a secondary branch if (n – i)/2 ≦ j ≦ n – 1 ≦ i; if has a primary branch B with i nodes and a secondary branch C with j nodes, then a branch D with h nodes is a tertiary branch if

Circumflex Accessory Branch of the Superficial Fibular Nerve: An Unusual Cause of Nerve Entrapment

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Patrick A. McEneaney ◽  
Joseph D. Rundell ◽  
Douglas P. Pacaccio ◽  
Thomas S. Nordquist
Keyword(s):  
Case Report ◽  
Peroneal Nerve ◽  
Surgical Excision ◽  
Lower Leg ◽  
Nerve Entrapment ◽  
Deep Fascia ◽  
Secondary Branch ◽  
Tertiary Branch ◽  
Fibular Nerve ◽  
Superficial Fibular Nerve

The superficial fibular (peroneal) nerve traditionally courses through the anterolateral deep leg and pierces the deep crural fascia at the lower leg to divide into its terminal branches. Entrapment of the superficial fibular nerve is most commonly documented to occur at where it pierces the deep fascia, and numerous etiologies causing entrapment are described. In this case report, we describe an unusual cause of entrapment from a tertiary branch of the superficial fibular nerve taking a circumflex course and wrapping around the secondary branch of the main nerve. This was successfully treated by surgical excision. To the best of our knowledge, this cause of entrapment has not been described in the literature at the time of this publication.

scholarly journals Maximal Subgroups and Irreducible Representations of Generalized Multi-Edge Spinal Groups

2018 ◽  
Vol 61 (3) ◽  
pp. 673-703 ◽  
Author(s):  
Benjamin Klopsch ◽  
Anitha Thillaisundaram
Keyword(s):  
Rooted Tree ◽  
Maximal Subgroups ◽  
Irreducible Representations ◽  
Prime Field ◽  
Infinite Dimensional ◽  
Special Cases ◽  
Profinite Completions ◽  
Spinal Group ◽  
Groups Acting On Trees ◽  
Image Position

AbstractLet p ≥ 3 be a prime. A generalized multi-edge spinal group $$G = \langle \{ a\} \cup \{ b_i^{(j)} {\rm \mid }1 \le j \le p,\, 1 \le i \le r_j\} \rangle \le {\rm Aut}(T)$$ is a subgroup of the automorphism group of a regular p-adic rooted tree T that is generated by one rooted automorphism a and p families $b^{(j)}_{1}, \ldots, b^{(j)}_{r_{j}}$ of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalizes the concepts of multi-edge spinal groups, including the widely studied GGS groups (named after Grigorchuk, Gupta and Sidki), and extended Gupta–Sidki groups that were introduced by Pervova [‘Profinite completions of some groups acting on trees, J. Algebra310 (2007), 858–879’]. Extending techniques that were developed in these more special cases, we prove: generalized multi-edge spinal groups that are torsion have no maximal subgroups of infinite index. Furthermore, we use tree enveloping algebras, which were introduced by Sidki [‘A primitive ring associated to a Burnside 3-group, J. London Math. Soc.55 (1997), 55–64’] and Bartholdi [‘Branch rings, thinned rings, tree enveloping rings, Israel J. Math.154 (2006), 93–139’], to show that certain generalized multi-edge spinal groups admit faithful infinite-dimensional irreducible representations over the prime field ℤ/pℤ.

scholarly journals Genetic analysis for panicle characters in diallel cross of rice

1970 ◽  
Vol 33 (4) ◽  
pp. 631-638
Author(s):  
KM Iftekharuddaula ◽  
MA Newaz ◽  
MA Salam ◽  
Khaleda Akter
Keyword(s):  
Genetic Analysis ◽  
Genetic Model ◽  
Branch Length ◽  
Diallel Cross ◽  
Primary Branch ◽  
Secondary Branch ◽  
Components Of Variance ◽  
Genetic Components ◽  
Recessive Genes ◽  
Dominant Genes

An experiment was carried out to study the genetic components for eight panicle characters in rice using an 8-parent half diallel cross excluding reciprocals during Transplant Aman season, 2003. The parental genotypes used in the study were BRRI dhan29, BR4828-54-4-l-4-9, BRR1 dhan28, 1R8, Amol3, 1R65610-38-2-4-2-6-3, Minikit and ZhongYu7, which were chosen for their diversity in panicle characters. Hayman's analysis of variance (ANOVA) indicated importance of both additive and non-additive genetic components for all the panicle characters except dominance component for filled grains/secondary branches. The ANOVA showed unidirectional dominance for the characters viz, primary branch length, secondary branch length, primary branches/panicle, secondary branches/panicle and filled grains/primary branch, asymmetrical gene distribution for all the panicle traits except filled grains/secondary branch and residual dominance effects for all the panicle characters studied. Two out of eight panicle characters viz., primary branches/panicle and unfilled grains/ secondary branch followed the simple additive-dominance genetic model. The rest of the panicle characters showed nonallelic gene interaction or epistasis. According to Vr-Wr graph, partial dominance was involved in the action of genes governing the inheritance of primary branches/panicle, while complete dominance was involved in the inheritance of unfilled grains/secondary branch. Most of the dominant genes for primary branches/panicle belonged to other hand, 1R8 possessed most of the dominant genes, while 1R65610-38-2-4-2-6-3 possessed most of the recessive genes for unfilled grains/secondary branch. The estimates of components of variance demonstrated involvement of both additive and dominant components in the inheritance of primary branches/panicle and unfilled grains/secondary branch. The distribution of dominant and recessive genes was unequal in the parents for these two characters also. There was drastic influence of environment on these two panicle characters following simple additive-dominance genetic model. Heritability in narrow sense (h2 ns) was very high for primary branches/panicle and unfilled grains/secondary branch. Key Words: Genetic analysis, diallel cross, panicle characters, rice. doi: 10.3329/bjar.v33i4.2307 Bangladesh J. Agril. Res. 33(4) : 631-638, December 2008

scholarly journals On Interpolation by Iteration of Proportional Parts, without the Use of Differences

1932 ◽  
Vol 3 (1) ◽  
pp. 56-76 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Divided Difference ◽  
Weighted Average ◽  
Unit Interval ◽  
Divided Differences ◽  
Linear Case ◽  
Not Given ◽  
Image Position ◽  
Fundamental Formula

Linear interpolation between two values of a function ua and ub can be performed, as is well known, in either of two ways. If the divided difference (ub−ua)/(b−a), which is usually denoted by u (a, b) or u (b, a), is provided, or its equivalent in tables at unit interval (the ordinary difference), we should generally prefer to use the formulawhich is the linear case of Newton's fundamental formula for interpolation by divided differences. If differences are not given, but a machine is available, then the use of proportional parts in the form of the weighted averagethe linear case of Lagrange's formula, is actually more convenient, since it involves no clearing of the product dials until the final result is read.

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 Growth Response of Two Pepper (Piper nigrum L.) Stem Cuttings on Application of IBA (Indole Butyric Acid) and NAA (Naphthalene Acetic Acid)

2018 ◽  
Vol 1 (1) ◽  
pp. 87-95
Author(s):  
Rizky Wulandari ◽  
Yaya Hasanah ◽  
Meiriani Meiriani
Keyword(s):  
Acetic Acid ◽  
Butyric Acid ◽  
Growth Response ◽  
Block Design ◽  
Primary Branch ◽  
Piper Nigrum ◽  
Naphthalene Acetic Acid ◽  
Secondary Branch ◽  
Indole Butyric Acid ◽  
Significant Difference

Using fruit branch for pepper shrub propagation is one of alternatives for an efficient pepper multiplication which usually uses underlayer cuttings. This research is aimed at finding the growth response of two pepper (Piper nigrum L.) cuttings to the administration of IBA (Indole Butyric Acid) and NAA (Naphthalene Acetic Acid). This research was conducted at the greenhouse of the Faculty of Agriculture, University of Sumatera Utara, Medan (± 32 m above sea level), from April to August 2017 using a factorial randomized block design with 2 factors, pepper branch cuttings (primary branch cuttings and secondary branch cuttings) and the administration of IBA and NAA (0+0 ppm, 2500+0 ppm, 0+2500 ppm, 1500+1000 ppm, and 1000+1500 ppm). The results show that the emerging shoot rate in the secondary branch cuttings was significantly faster than in the primary branch cuttings, but the volume of root in the primary branch cuttings is significantly larger than in the secondary branch cuttings. There was no significant difference in the administration of IBA and NAA on all observed variables. The highest interaction of shoot length was found in the  primary branch cuttings with the administration of  IBA 1500 ppm + NAA 1000 ppm and the highest percentage of root and root volume was found in the primary branch cuttings with the administration of IBA 2500 ppm + NAA 0 ppm.

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.

scholarly journals Cutting down random trees

1970 ◽  
Vol 11 (3) ◽  
pp. 313-324 ◽  
Author(s):  
A. Meir ◽  
J. W. Moon
Keyword(s):  
Original Problem ◽  
Expected Value ◽  
Random Trees ◽  
Rooted Trees ◽  
Not Given

Let Tn denote a tree with n(≧ 2) labelled points: we assume Tn is rooted at a given point x, say the point labelled 1 (see [3] for definitions not given here). If we remove some edge e of Tn, then Tn falls into two subtrees one of which, say Tk, contains the root x. If k ≧ 2 we can remove some edge of Tk and obtain an even smaller subtree of Tn that contains x. If we repeat this process we will eventually obtain the subtree consisting of x itself. Let λ = λ(Tn) denote the number of edges removed from Tn before the root x is isolated. Our main object here is to determine the expected value μ(n) and variance σ2(n) of λ(Tn) under the assumptions (1) Tn is chosen at random from the set of nn−2 trees with n labelled points that are rooted at point x, and (2) at each stage the edge removed is chosen at random from the edges of the remaining subree containing x. It follows from our results that μ(n) ~ (½πn)½ and (2−½π)n ~ (2−½π)n as n tends to infinity. We also consider the corresponding problem for forests of rooted trees and for trees in which the degree of the root is specified. We are indebted to Professor Alistair Lachlan for suggesting the original problem to us.

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.

Tenuibiotus, a new genus of Macrobiotidae (Eutardigrada)

Zootaxa ◽  
2011 ◽  
Vol 2761 (1) ◽  
pp. 34 ◽  
Author(s):  
GIOVANNI PILATO ◽  
OSCAR LISI
Keyword(s):  
New Genus ◽  
Buccal Cavity ◽  
Primary Branch ◽  
Secondary Branch ◽  
Long Distance ◽  
The Common

The group of Macrobiotus species having claws of the tenuis-type is discussed and a new genus, named Tenuibiotus, is instituted. The tenuis-type claws are characterised by the fusion of the primary and secondary branches over a long distance so that the common tract is longer than that found in other members of the Macrobiotidae. Distally, the secondary branch is clearly shorter than the primary branch, and forms with it almost a right angle. In all the known species attributable to the new genus, the buccal tube is narrow, the buccal cavity is small with small peribuccal lamellae; the eggs are laid freely and have conical or trunco-conical processes.

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