Статті в журналах: "Rooted tree"

1

Tanaka, Tatsushi. "Rooted tree maps." Communications in Number Theory and Physics 13, no. 3 (2019): 647–66. http://dx.doi.org/10.4310/cntp.2019.v13.n3.a6.

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2

Béal, Sylvain, Eric Rémila, and Philippe Solal. "Rooted-tree solutions for tree games." European Journal of Operational Research 203, no. 2 (June 2010): 404–8. http://dx.doi.org/10.1016/j.ejor.2009.07.023.

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3

Fumuro, Masahiko. "Comparison of Growth, Yield, and Fruit Quality of Pot-planted Mango cv. Aikou Using Own-rooted Trees Propagated by Air Layering and Grafted Trees Propagated by Conventional Methods." HortScience 54, no. 7 (July 2019): 1175–80. http://dx.doi.org/10.21273/hortsci13984-19.

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To determine the potential of using own-rooted trees to lower tree height and delay the decline in tree vigor caused by root clogging, the growth, yield, and fruit quality of pot-planted ‘Aikou’ mango (Mangifera indica L.) trees propagated by air layering and grafting were observed for 8 years after planting. The trunk diameter of the own-rooted trees propagated by air layering (hereafter abbreviated as own-rooted trees) was significantly smaller than that of the grafted trees propagated by conventional methods (hereafter abbreviated as grafted trees), but there were no significant differences in the scion diameters of the grafted trees. Moreover, no significant differences were observed in leaf number or total length of green branches between the own-rooted and grafted trees during the final 3 years. The height of the own-rooted trees was significantly shorter than that of the grafted trees. Although no difference in the fresh or dry weight of the aboveground part and whole tree was observed between the own-rooted and grafted trees, the fresh and dry weights of the underground part of the own-rooted trees were significantly lower than those of the grafted trees. Furthermore, the T-R ratio (the weight of the aboveground part of the tree excluding the leaves/the weight of the underground part of the tree) of the own-rooted trees was significantly higher than that of the grafted trees. Overall, no significant differences in yield or fruit quality were observed between the two tree types, and the average yield per 1 m2 over 6 years was 2.9–3.1 kg. These results indicate that it may be possible to lower tree height, delay the decline in tree vigor caused by root clogging, and prolong the life span of pot-planted trees by using own-rooted trees.

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4

GAWRON, PIOTR W., VOLODYMYR V. NEKRASHEVYCH, and VITALY I. SUSHCHANSKY. "CONJUGATION IN TREE AUTOMORPHISM GROUPS." International Journal of Algebra and Computation 11, no. 05 (October 2001): 529–47. http://dx.doi.org/10.1142/s021819670100070x.

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It is given a full description of conjugacy classes in the automorphism group of the locally finite tree and of a rooted tree. They are characterized by their types (a labeled rooted trees) similar to the cyclical types of permutations. We discuss separately the case of a level homogenous tree, i.e. conjugality in wreath products of infinite sequences of symmetric groups. It is proved those automorphism groups of rooted and homogenous non-rooted trees are ambivalent.

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5

Mastakas, Konstantinos. "Drawing a rooted tree as a rooted y−monotone minimum spanning tree." Information Processing Letters 166 (February 2021): 106035. http://dx.doi.org/10.1016/j.ipl.2020.106035.

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6

HE, YING-JUN, TRINH N. D. HUYNH, JESPER JANSSON, and WING-KIN SUNG. "INFERRING PHYLOGENETIC RELATIONSHIPS AVOIDING FORBIDDEN ROOTED TRIPLETS." Journal of Bioinformatics and Computational Biology 04, no. 01 (February 2006): 59–74. http://dx.doi.org/10.1142/s0219720006001709.

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To construct a phylogenetic tree or phylogenetic network for describing the evolutionary history of a set of species is a well-studied problem in computational biology. One previously proposed method to infer a phylogenetic tree/network for a large set of species is by merging a collection of known smaller phylogenetic trees on overlapping sets of species so that no (or as little as possible) branching information is lost. However, little work has been done so far on inferring a phylogenetic tree/network from a specified set of trees when in addition, certain evolutionary relationships among the species are known to be highly unlikely. In this paper, we consider the problem of constructing a phylogenetic tree/network which is consistent with all of the rooted triplets in a given set [Formula: see text] and none of the rooted triplets in another given set [Formula: see text]. Although NP-hard in the general case, we provide some efficient exact and approximation algorithms for a number of biologically meaningful variants of the problem.

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7

Lu, Pengxin, Wayne Bell, Paul Charrette, and Megan Thompson. "Performance of jack pine (Pinus banksiana) rooted cuttings from proliferated dwarf shoots versus seedlings 8 years after planting." Canadian Journal of Forest Research 42, no. 7 (July 2012): 1404–9. http://dx.doi.org/10.1139/x2012-079.

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Growth and tree form characteristics of jack pine (Pinus banksiana Lamb.) rooted cuttings propagated from proliferated dwarf shoots (PDS) were compared with seedlings in two field trials 8 years after establishment. Results indicated that jack pine rooted cuttings from PDS can grow as well as seedlings and maintain acceptable tree form. Rooted cuttings of progeny from the 22 top-ranking open-pollinated families in a seedling seed orchard of jack pine were 4.2% taller and 10% larger in diameter at breast height than commercial seedlings tested on the same sites, which indicates that rooted cuttings have potential in realizing genetic gains in jack pine tree improvement programs. Rooted cuttings increased the proportion of trees with normal branching characteristics and reduced the percentage of trees with excessive heavy branches in the Sault St. Marie trial, which had larger tree sizes. However, longer term monitoring (20 to 25 years) is needed to determine stability of jack pine rooted cuttings planted on sandy soil where wind throw may become a problem as tree size increases.

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8

Abdulcarim, Normalah Sharief, and Susan C. Dagondon. "On the Independent Neighborhood Polynomial of the Rooted Product of Two Trees." European Journal of Pure and Applied Mathematics 15, no. 1 (January 31, 2022): 64–81. http://dx.doi.org/10.29020/nybg.ejpam.v15i1.4220.

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Let G be a connected graph. We say that a given graph is a tree if every pair ofvertices is connected by a unique path. The rooted product of two trees is relevant to tree, as the obtained product is another tree. In this paper, we establish the independent neighborhood sets of a tree and obtain its corresponding independent neighborhood polynomial. Furthermore, the independent neighborhood polynomial of the rooted product of two trees were determine using their independent neighborhood sets.

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9

Tsipouridis, C. G., A. Isaakidis, A. Manganaris, I. Therios, and Z. Michailidis. "Propagation and field performance of own-rooted peach trees." Australian Journal of Experimental Agriculture 44, no. 12 (2004): 1225. http://dx.doi.org/10.1071/ea02062.

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Ten peach and nectarine (Prunus persica L. Batsh) cultivars: Arm King, Early Crest, Early Gem, Flavor Crest, May Crest, May Grand, Red Gold, Red Haven, Spring Crest and Sun Crest, were propagated by both hardwood cuttings (HC) and by bud grafting onto peach seedlings. Significant differences were observed for rooting among cultivars and applied IBA. Degree of blooming and yield were higher for HC propagated own-rooted trees when compared with budded trees in the first 6 years of fruiting. Budded trees increased in size faster than HC trees but were less productive. Yield, yield efficiency and fruit size were not only cultivar specific, but were also affected by the propagation method, being higher for own-rooted trees in most cultivars. Tree mortality was generally higher for budded trees. No significant differences were found in mineral absorption efficiency, time of blooming, fruit firmness, acidity and sugar level between own-rooted and budded trees. Results based on percent rooting of HC, yield, fruit size, growth and tree mortality suggest that own-rooted HC trees should be an acceptable tree type for commercial orchards, especially for the cultivars Sun Crest, Spring Crest and Red Haven.

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10

Takacs, Lajos. "On the total heights of random rooted trees." Journal of Applied Probability 29, no. 3 (September 1992): 543–56. http://dx.doi.org/10.2307/3214892.

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Denote by Sn the set of all distinct rooted trees with n labeled vertices. Define τn as the total height of a tree chosen at random in the set Sn, assuming that all the possible nn–1 choices are equally probable. The total height of a tree is defined as the sum of the heights of its vertices. The height of a vertex in a rooted tree is the distance from the vertex to the root of the tree, that is, the number of edges in the path from the vertex to the root. This paper is concerned with the distribution and the moments of τn and their asymptotic behavior as n → ∞.

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11

Takacs, Lajos. "On the total heights of random rooted trees." Journal of Applied Probability 29, no. 03 (September 1992): 543–56. http://dx.doi.org/10.1017/s0021900200043370.

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Denote by Sn the set of all distinct rooted trees with n labeled vertices. Define τn as the total height of a tree chosen at random in the set Sn , assuming that all the possible nn –1 choices are equally probable. The total height of a tree is defined as the sum of the heights of its vertices. The height of a vertex in a rooted tree is the distance from the vertex to the root of the tree, that is, the number of edges in the path from the vertex to the root. This paper is concerned with the distribution and the moments of τn and their asymptotic behavior as n → ∞.

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12

Alanzi, Ayed A. R., and James H. Degnan. "Statistical inconsistency of the unrooted minimize deep coalescence criterion." PLOS ONE 16, no. 5 (May 10, 2021): e0251107. http://dx.doi.org/10.1371/journal.pone.0251107.

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Species trees, which describe the evolutionary relationships between species, are often inferred from gene trees, which describe the ancestral relationships between sequences sampled at different loci from the species of interest. A common approach to inferring species trees from gene trees is motivated by supposing that gene tree variation is due to incomplete lineage sorting, also known as deep coalescence. One of the earliest methods motivated by deep coalescence is to find the species tree that minimizes the number of deep coalescent events needed to explain discrepancies between the species tree and input gene trees. This minimize deep coalescence (MDC) criterion can be applied in both rooted and unrooted settings. where either rooted or unrooted gene trees can be used to infer a rooted species tree. Previous work has shown that MDC is statistically inconsistent in the rooted setting, meaning that under a probabilistic model for deep coalescence, the multispecies coalescent, for some species trees, increasing the number of input gene trees does not make the method more likely to return a correct species tree. Here, we obtain analogous results in the unrooted setting, showing conditions leading to inconsistency of the MDC criterion using the multispecies coalescent model with unrooted gene trees for four taxa and five taxa.

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13

Bartha, Dénes. "Reconstruction of Rooted Directed Trees." Acta Cybernetica 24, no. 2 (November 3, 2019): 249–62. http://dx.doi.org/10.14232/actacyb.24.2.2019.5.

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Let T be a rooted directed tree on n vertices, rooted at v. The rooted subtree frequency vector (RSTF-vector) of T with root v, denoted by rstf(T, v) is a vector of length n whose entry at position k is the number of subtrees of T that contain v and have exactly k vertices. In this paper we present an algorithm for reconstructing rooted directed trees from their rooted subtree frequencies (up to isomorphism). We show that there are examples of nonisomorphic pairs of rooted directed trees that are RSTF-equivalent, s.t. they share the same rooted subtree frequency vectors. We have found all such pairs (groups) for small sizes by using exhaustive computer search. We show that infinitely many nonisomorphic RSTF-equivalent pairs of trees exist by constructing infinite families of examples.

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14

Nagamochi, Hiroshi, and Kohei Okada. "Approximating the minmax rooted-tree cover in a tree." Information Processing Letters 104, no. 5 (November 2007): 173–78. http://dx.doi.org/10.1016/j.ipl.2007.06.012.

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15

Lyons, Jacqueline. "Rooted and Reaching: Vrksasana , Tree Pose." Fourth Genre: Explorations in Nonfiction 15, no. 2 (2013): 91–100. http://dx.doi.org/10.1353/fge.2013.0507.

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16

Provine, Robert C., Paek Tae-ung, Kim Hae-suk, Im Chae-Won, Song Hye-jin, and Paek Tae-ung. "The Deep-Rooted Tree Sanjo Collection." Ethnomusicology 35, no. 1 (1991): 160. http://dx.doi.org/10.2307/852414.

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17

Kinser, Ryan. "Rank functions on rooted tree quivers." Duke Mathematical Journal 152, no. 1 (March 2010): 27–92. http://dx.doi.org/10.1215/00127094-2010-006.

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18

Deutsch, Emeric. "Rooted tree statistics from Matula numbers." Discrete Applied Mathematics 160, no. 15 (October 2012): 2314–22. http://dx.doi.org/10.1016/j.dam.2012.05.012.

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19

TANAKA, EIICHI. "A NOTE ON A TREE-TO-TREE EDITING PROBLEM." International Journal of Pattern Recognition and Artificial Intelligence 09, no. 01 (February 1995): 167–72. http://dx.doi.org/10.1142/s0218001495000092.

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In the previous paper on a tree-to-tree editing problem some errors were included. This letter describes the corrected definition of a structure preserving mapping between rooted and ordered trees and a computing method of the tree distance based on the mapping.

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20

Hammerschlag, F. A., and R. Scorza. "Field Performance of Micropropagated, Own-rooted Peach Trees." Journal of the American Society for Horticultural Science 116, no. 6 (November 1991): 1089–91. http://dx.doi.org/10.21273/jashs.116.6.1089.

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Four peach [Prunus persica (L.) Batsch] scion cultivars, `Jerseyqueen', `Redskin', `Suncrest', and `Sunhigh', that were propagated by tissue culture techniques and by bud-grafting onto `Lovell' seedlings, were compared at Kearneysville, W.Va., and at Beltsville, Md. At Kearneysville, total fruit production was higher for tissue-cultured (TC) trees when compared with budded trees in the first 3 years of fruiting, whereas trunk diameter increases were generally larger for budded trees. In the following year, fruit production was similar for both TC and budded trees, although trunk diameter increases continued to be larger for budded trees. At Beltsville, fruit production was significantly higher for TC trees in 1987, the first fruiting season, but the same for both in the second season. Trunk diameter increases were larger for budded trees both years. Differences in tree growth and productivity in the early years of orchard establishment appeared to be related to the size of plants that were planted. Budded trees, which were smaller than TC trees at planting, increased in size faster than TC trees but were less productive. Crop efficiency was cultivar-specific, but differences among cultivars was less if trees were TC propagated. These results suggested that based on yield and growth, own-rooted TC trees should be an acceptable tree type for commercial orchards.

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21

Kubicka, Ewa. "An Efficient Method of Examining all Trees." Combinatorics, Probability and Computing 5, no. 4 (December 1996): 403–13. http://dx.doi.org/10.1017/s0963548300002157.

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In this paper, we present a techique for examining all trees of a given order. Our approach is based on the Beyer and Hedetniemi algorithm for generating all rooted trees of a given order and on the Wright, Richmond, Odlyzko and McKay algorithm for generating all free trees of a given order. In the introduction we describe these algorithms. We also give a precise evaluation of the average number of moves it takes to generate a rooted tree, which improves the upper bound given by Beyer and Hedetniemi. In the second section we present a new method of examining all trees which uses these generating algorithms. The last section contains two applications of the method introduced. The main result of the paper is that the average number of steps required by the proposed algorithm to examine a rooted tree is bounded by a constant independent of the order of a tree.

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22

Raman, Indhumathi. "On Counting and Embedding a Subclass of Height-Balanced Trees." Modelling and Simulation in Engineering 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/748941.

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A height-balanced tree is a rooted binary tree in which, for every vertexv, the difference in the heights of the subtrees rooted at the left and right child ofv(called the balance factor ofv) is at most one. In this paper, we consider height-balanced trees in which the balance factor of every vertex beyond a level is0. We prove that there are22t-1such trees and embed them into ageneralized join of hypercubes.

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23

Fins, Lauren, and Verna Reedy. "Cone Production by Rooted Cuttings, Grafts, and Seedlings of Western Larch." Western Journal of Applied Forestry 7, no. 4 (October 1, 1992): 108–9. http://dx.doi.org/10.1093/wjaf/7.4.108.

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Abstract Cone production in 1991 was compared among rooted cuttings from juvenile trees, grafts from mature scions, and seedlings planted in a common garden in Plains, MT, in 1981 and 1983. The differences among tree types were statistically significant for the mean number of cones per tree, but were not significant for the percent of trees producing cones. A projection of cone production per 1,000 trees showed that the grafts would produce nearly twice the number of cones as the seedlings and more than five times the number of cones as the rooted cuttings under the same conditions. West. J. Appl. For. 7(4):108-109.

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24

Nathanson, Melvyn B. "A forest of linear fractional transformations." International Journal of Number Theory 11, no. 04 (April 29, 2015): 1275–99. http://dx.doi.org/10.1142/s1793042115500694.

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The Calkin–Wilf tree is a rooted infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form a/b, where a and b are relatively prime positive integers. For every 2 × 2 matrix with nonnegative integral coordinates and nonzero determinant, it is possible to construct an analogous tree with this root. If the root is the identity matrix, then the tree consists of all matrices with determinant 1, and this tree possesses the basic properties of the Calkin–Wilf tree of positive rational numbers. The set of all matrices with nonnegative integral coordinates and nonzero determinant decomposes into a forest of rooted infinite binary trees.

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25

Cheng, Zhiyun, Sujoy Mukherjee, Józef H. Przytycki, Xiao Wang, and Seung Yeop Yang. "Realization of plucking polynomials." Journal of Knot Theory and Its Ramifications 26, no. 02 (February 2017): 1740016. http://dx.doi.org/10.1142/s0218216517400168.

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26

Hu, Zhishui, Zheng Li, and Qunqiang Feng. "Accessibility percolation on random rooted labeled trees." Journal of Applied Probability 56, no. 2 (June 2019): 533–45. http://dx.doi.org/10.1017/jpr.2019.29.

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AbstractThe accessibility percolation model is investigated on random rooted labeled trees. More precisely, the number of accessible leaves (i.e. increasing paths) Zn and the number of accessible vertices Cn in a random rooted labeled tree of size n are jointly considered in this work. As n → ∞, we prove that (Zn, Cn) converges in distribution to a random vector whose probability generating function is given in an explicit form. In particular, we obtain that the asymptotic distributions of Zn + 1 and Cn are geometric distributions with parameters e/(1 + e) and 1/e, respectively. Much of our analysis is performed in the context of local weak convergence of random rooted labeled trees.

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27

Chen, Qizhao, and Christina Goldschmidt. "Parking on a random rooted plane tree." Bernoulli 27, no. 1 (February 2021): 93–106. http://dx.doi.org/10.3150/20-bej1227.

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28

Andriantiana, Eric O. D., Kenneth Dadedzi, and Stephan Wagner. "The ancestral matrix of a rooted tree." Linear Algebra and its Applications 575 (August 2019): 35–65. http://dx.doi.org/10.1016/j.laa.2019.04.004.

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29

Szwarcfiter, Jayme L. "On digraphs with a rooted tree structure." Networks 15, no. 1 (1985): 49–57. http://dx.doi.org/10.1002/net.3230150106.

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30

Moore, N. C. A., and P. Prosser. "The Ultrametric Constraint and its Application to Phylogenetics." Journal of Artificial Intelligence Research 32 (August 28, 2008): 901–38. http://dx.doi.org/10.1613/jair.2580.

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A phylogenetic tree shows the evolutionary relationships among species. Internal nodes of the tree represent speciation events and leaf nodes correspond to species. A goal of phylogenetics is to combine such trees into larger trees, called supertrees, whilst respecting the relationships in the original trees. A rooted tree exhibits an ultrametric property; that is, for any three leaves of the tree it must be that one pair has a deeper most recent common ancestor than the other pairs, or that all three have the same most recent common ancestor. This inspires a constraint programming encoding for rooted trees. We present an efficient constraint that enforces the ultrametric property over a symmetric array of constrained integer variables, with the inevitable property that the lower bounds of any three variables are mutually supportive. We show that this allows an efficient constraint-based solution to the supertree construction problem. We demonstrate that the versatility of constraint programming can be exploited to allow solutions to variants of the supertree construction problem.

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31

Bachmann, Henrik, and Tatsushi Tanaka. "Rooted tree maps and the derivation relation for multiple zeta values." International Journal of Number Theory 14, no. 10 (October 25, 2018): 2657–62. http://dx.doi.org/10.1142/s1793042118501592.

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Rooted tree maps assign to an element of the Connes–Kreimer Hopf algebra of rooted trees a linear map on the noncommutative polynomial algebra in two letters. Evaluated at any admissible word, these maps induce linear relations between multiple zeta values. In this note, we show that the derivation relations for multiple zeta values are contained in this class of linear relations.

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32

Mayer, Newton Alex, Renan Navroski, Jorge Atílio Benati, Fernanda Maísa Roth, Bernardo Ueno, Gilberto Nava, and Luis Eduardo Corrêa Antunes. "Leaf nutrient content of ‘Jade’ peach grafted on 22 clonal rootstock and in own-rooted trees." Comunicata Scientiae 14 (August 21, 2023): e3983. http://dx.doi.org/10.14295/cs.v14.3983.

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The objective of this research was to evaluate the effects of 22 clonal rootstocks and own-rooted scion trees (without rootstock) on leaf nutrient contents (N, P, K, Ca, Mg, Mn, Fe, Zn, Cu and B) of the ‘Jade’ scion peach growing in a no-irrigated area with Peach Tree Short Life (PTSL) history, as well as their effects on nutrient agronomic interpretation. Macronutrient (N, P, K, Ca and Mg) and micronutrient contents (B, Cu, Fe, Mn and Zn) were determined in the first and second years after tree planting, in Pelotas-RS, Brazil. We conclude that leaf contents of P, K, Ca, Mg, B, Cu, Mn and Zn are influenced by the scion/rootstock combinations and own-rooted trees tested. Treatments changed agronomic interpretation classes of all macro and micronutrients. For macronutrients, ‘Flordaguard’, De Guia, Tardio-01 rootstocks and the own-rooted trees stood out, with leaf nutrient contents similar or even higher than trees grafted on ‘Capdeboscq’ and ‘Aldrighi’. For micronutrients, trees on GxN.9, ‘Ishtara’ and ‘Santa Rosa’ plum stood out. From a nutritional point of view, own-rooted ‘Jade’ peach trees did not present any limitations.

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33

Ando, Kazutoshi, and Shinji Kato. "REDUCTION OF ULTRAMETRIC MINIMUM COST SPANNING TREE GAMES TO COST ALLOCATION GAMES ON ROOTED TREES." Journal of the Operations Research Society of Japan 53, no. 1 (2010): 62–68. http://dx.doi.org/10.15807/jorsj.53.62.

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34

Zhou, Chenping, Ruiting Chen, Yaqiang Sun, He Wang, Yi Wang, Ting Wu, Xinzhong Zhang, Xuefeng Xu, and Zhenhai Han. "Effect of Bridge Grafting the M9 Self-rooted Rootstock in Trunk-wounded Apple Trees on Vegetative Growth, Yield, and Fruit Characteristics." HortScience 53, no. 7 (July 2018): 937–45. http://dx.doi.org/10.21273/hortsci13122-18.

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Bridge grafting is widely applied in trunk-wounded apple trees. In this study, we carried out semigirdling and ring girdling on the trunk of ‘Nagafu 2’/Malus baccata (L.) Borkh apple trees to simulate trunk injury. We then bridge grafted a M9 self-rooted rootstock on the injured trunks to study the effects of bridge grafting on flowering, fruit-set, tree vigor, and fruit characteristics in ‘Nagafu 2’ apple. The results showed that both semigirdling and ring girdling due to the large wounded area caused significant decrease in flowering, fruit-set, and tree vigor (estimated by measuring leaf area, leaf gas exchange, tree height, and shoot growth); in addition, ring girdling increased flesh and peel firmness. However, bridge grafting of M9 self-rooted rootstock on semigirdling and girdling apple trees resulted in partial recovery of tree vigor (leaf area and photosynthesis) and maintaining the reduction of vegetative growth, thereby increasing flowering, fruit-set, yield, fruit weight, and peel firmness.

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35

Hiesmayr, Ella, and Ümit Işlak. "Asymptotic results on Hoppe trees and their variations." Journal of Applied Probability 57, no. 2 (June 2020): 441–57. http://dx.doi.org/10.1017/jpr.2020.12.

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AbstractA uniform recursive tree on n vertices is a random tree where each possible $(n-1)!$ labelled recursive rooted tree is selected with equal probability. We introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees, providing diversity among the nodes and making the model more flexible for applications. We analyse the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weighted trees.

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36

D’Angeli, Daniele, and Alfredo Donno. "Structure Polynomials and Subgraphs of Rooted Regular Trees." Algebra Colloquium 25, no. 01 (January 22, 2018): 45–70. http://dx.doi.org/10.1142/s1005386718000044.

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We introduce an algebraic structure allowing us to describe subgraphs of a regular rooted tree. Its elements are called structure polynomials, and they are in a one- to-one correspondence with the set of all subgraphs of the tree. We define two operations, the sum and the product of structure polynomials, giving a graph interpretation of them. Then we introduce an equivalence relation between polynomials, using the action of the full automorphism group of the tree, and we count equivalence classes of subgraphs modulo this equivalence. We also prove that this action gives rise to symmetric Gelfand pairs. Finally, when the regularity degree of the tree is a prime p, we regard each level of the tree as a finite dimensional vector space over the finite field 𝔽p, and we are able to completely characterize structure polynomials corresponding to subgraphs whose leaf set is a vector subspace.

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37

KUMAR, Dr D. SURESH. "THE BAOBAB TREE." Hygeia J. D.Med.10 (1) August 2018 - January 2019 10, no. 1 (September 15, 2018): 1–2. http://dx.doi.org/10.15254/h.j.d.med.10.2018.16.

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Trees are known to live for many years. Gautama Buddha attained enlightenment while meditating underneath a peepal tree (Ficus religiosa). A branch of the original tree was rooted in Anuradhapura, Sri Lanka in 288 B.C. and is known as Jaya Sri Maha Bodhi. It is the oldest plant in the world. Long-living plants are found in many parts of the world. The Baobab tree is one among them. Baobab is the common name of a genus of trees (Adansonia) distributed in Madagascar, Africa, Australia and India. The Baobab is the national tree of Madagascar. The Baobab is also known as “bottle tree”, “the tree of life”, “upside-down tree”, and “monkey bread tree”.

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38

Nakahara, Yuta, Shota Saito, Akira Kamatsuka, and Toshiyasu Matsushima. "Probability Distribution on Full Rooted Trees." Entropy 24, no. 3 (February 24, 2022): 328. http://dx.doi.org/10.3390/e24030328.

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The recursive and hierarchical structure of full rooted trees is applicable to statistical models in various fields, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is not a random variable; as such, model selection to avoid overfitting is problematic. One method to solve this problem is to assume a prior distribution on the full rooted trees. This enables the optimal model selection based on Bayes decision theory. For example, by assigning a low prior probability to a complex model, the maximum a posteriori estimator prevents the selection of the complex one. Furthermore, we can average all the models weighted by their posteriors. In this paper, we propose a probability distribution on a set of full rooted trees. Its parametric representation is suitable for calculating the properties of our distribution using recursive functions, such as the mode, expectation, and posterior distribution. Although such distributions have been proposed in previous studies, they are only applicable to specific applications. Therefore, we extract their mathematically essential components and derive new generalized methods to calculate the expectation, posterior distribution, etc.

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39

Kappel, Frank, and Michel Bouthillier. "Rootstock, severity of dormant pruning, and summer pruning influences on peach tree size, yield, and fruit quality." Canadian Journal of Plant Science 75, no. 2 (April 1, 1995): 491–96. http://dx.doi.org/10.4141/cjps95-086.

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Peach trees in British Columbia are pruned heavily during the dormant season with some potentially detrimental effects. Summer pruning has been used to control fruit tree vigor and improve fruit quality. The objective of this study was to reduce tree size, increase yield, and improve fruit color by using a less severe dormant pruning system and summer pruning prior to harvest. Over 4 yr, mature, self-rooted (micropropagated) Fairhaven peach trees and Fairhaven on Siberian C rootstock were subjected to two different dormant pruning regimes, with or without summer pruning. Yield, tree growth, pruning weights (dormant and summer) and fruit quality (size and color) measurements were recorded annually. There were no differences in yields per tree for the two rootstocks or for the summer pruning treatments. The lighter ("long") dormant pruning increased yields but average fruit weight was higher in the heavier ("short") dormant pruning treatment. Summer pruning increased the amount of red color on the fruit but only slightly. Own-rooted trees were larger (tree height and ground area covered) than trees budded on Siberian C. Tree height was also increased by the lighter dormant pruning treatment. The partitioning index was higher for trees on Siberian C rootstock, "long" dormant pruned, or non- summer pruned trees. Key words:Prunus persica, fruit color, fruit size, partitioning index

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40

Coleman, Gareth A., Adrián A. Davín, Tara A. Mahendrarajah, Lénárd L. Szánthó, Anja Spang, Philip Hugenholtz, Gergely J. Szöllősi, and Tom A. Williams. "A rooted phylogeny resolves early bacterial evolution." Science 372, no. 6542 (May 6, 2021): eabe0511. http://dx.doi.org/10.1126/science.abe0511.

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A rooted bacterial tree is necessary to understand early evolution, but the position of the root is contested. Here, we model the evolution of 11,272 gene families to identify the root, extent of horizontal gene transfer (HGT), and the nature of the last bacterial common ancestor (LBCA). Our analyses root the tree between the major clades Terrabacteria and Gracilicutes and suggest that LBCA was a free-living flagellated, rod-shaped double-membraned organism. Contrary to recent proposals, our analyses reject a basal placement of the Candidate Phyla Radiation, which instead branches sister to Chloroflexota within Terrabacteria. While most gene families (92%) have evidence of HGT, overall, two-thirds of gene transmissions have been vertical, suggesting that a rooted tree provides a meaningful frame of reference for interpreting bacterial evolution.

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41

Kochubinska, Eugenia. "Spectrum of partial automorphisms of regular rooted tree." Semigroup Forum 103, no. 2 (August 6, 2021): 567–74. http://dx.doi.org/10.1007/s00233-021-10219-5.

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42

PETIT, FRANCK, and VINCENT VILLAIN. "OPTIMALITY AND SELF-STABILIZATION IN ROOTED TREE NETWORKS." Parallel Processing Letters 10, no. 01 (March 2000): 3–14. http://dx.doi.org/10.1142/s0129626400000032.

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In this paper, we consider arbitrary tree networks where every processor, except one, called the root, executes the same program. We show that, to design a depth-first token circulation protocol in such networks, it is necessary to have at least [Formula: see text] configurations, where n is the number of processors in the network and Δi is the degree of processor pi. We then propose a depth-first token circulation algorithm which matches the above minimal number of configurations. We show that the proposed algorithm is self-stabilizing, i.e., the system eventually recovers itself to a legitimate state after any perturbation modifying the state of the processors. Hence, the proposed algorithm is optimal in terms of the number of configurations and no extra cost is involved in making it stabilizing.

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43

PETIT, FRANCK, and VINCENT VILLAIN. "OPTIMALITY AND SELF-STABILIZATION IN ROOTED TREE NETWORKS." Parallel Processing Letters 09, no. 03 (September 1999): 313–23. http://dx.doi.org/10.1142/s0129626499000293.

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In this paper, we consider arbitrary tree networks where every processor, except one, called the root, executes the same program. We show that, to design a depth-first token circulation protocol in such networks, it is necessary to have at least [Formula: see text] configurations, where n is the number of processors in the network and Δi is the degree of processor pi. We then propose a depth-first token circulation algorithm which matches the above minimal number of configurations. We show that the proposed algorithm is self-stabilizing, i.e., the system eventually recovers itself to a legitimate state after any perturbation modifying the state of the processors. Hence, the proposed algorithm is optimal in terms of the number of configurations and no extra cost is involved in making it stabilizing.

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44

Jacqueline Lyons. "Rooted and Reaching: Vrksasana, Tree Pose." Fourth Genre: Explorations in Nonfiction 15, no. 2 (2013): 91. http://dx.doi.org/10.14321/fourthgenre.15.2.0091.

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45

Sefidgaran, Milad, and Aslan Tchamkerten. "Distributed Function Computation Over a Rooted Directed Tree." IEEE Transactions on Information Theory 62, no. 12 (December 2016): 7135–52. http://dx.doi.org/10.1109/tit.2016.2530398.

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46

Wu, Bang Ye. "Constructing the Maximum Consensus Tree from Rooted Triples." Journal of Combinatorial Optimization 8, no. 1 (March 2004): 29–39. http://dx.doi.org/10.1023/b:joco.0000021936.04215.68.

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47

Tyomkyn, Mykhaylo. "A proof of the rooted tree alternative conjecture." Discrete Mathematics 309, no. 20 (October 2009): 5963–67. http://dx.doi.org/10.1016/j.disc.2009.04.025.

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48

GARG, ASHIM, MICHAEL T. GOODRICH, and ROBERTO TAMASSIA. "PLANAR UPWARD TREE DRAWINGS WITH OPTIMAL AREA." International Journal of Computational Geometry & Applications 06, no. 03 (September 1996): 333–56. http://dx.doi.org/10.1142/s0218195996000228.

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Rooted trees are usually drawn planar and upward, i.e., without crossings and with-out any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide linear-time algorithms for constructing optimal area drawings. Let T be a bounded-degree rooted tree with N nodes. Our results are summarized as follows: • We show that T admits a planar polyline upward grid drawing with area O(N), and with width O(Nα) for any prespecified constant a such that 0<α<1. • If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O (N log log N). • We show that if T is ordered, it admits an O(N log N)-area planar upward grid drawing that preserves the left-to-right ordering of the children of each node. • We show that all of the above area bounds are asymptotically optimal in the worst case. • We present O(N)-time algorithms for constructing each of the above types of drawings of T with asymptotically optimal area. • We report on the experimentation of our algorithm for constructing planar polyline upward grid drawings, performed on trees with up to 24 million nodes.

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49

Masson, Aurélien, and Olivier Monteuuis. "RUBBER TREE CLONAL PLANTATIONS: GRAFTED VS SELF-ROOTED PLANT MATERIAL." BOIS & FORETS DES TROPIQUES 332 (September 18, 2017): 57–68. http://dx.doi.org/10.19182/bft2017.332.a31333.

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The forest tree species Hevea brasiliensis is extensively planted in the humid tropics to meet the increasing demand for natural rubber. Huge quantities of planting stock are therefore needed. The seed option remains the easiest and cheapest way to establish plantations of rubber trees but those show a great variability for vigor and also for latex yield. The rationale of produ- cing clones for overcoming this variability was already obvious in the early 1910’s but due to the difficulties encountered at that time for rooting shoots, grafting was used as an alternative cloning method. The striking increase in yield noticed from the graft-derived clonal plantations warranted their large scale development. Eventually, the budded clones by virtue of their much higher and uniform produc- tivity supplanted the seedlings in most industrial plantations. However, grafting is also associated with drawbacks and for decades efforts aiming at mass producing selected rubber tree clones on their own roots by rooted cuttings have been pur- sued. However, this approach was pro- gressively abandoned due to disappoin- ting rooting results and, from the 70’s onwards, priority has been given to in vitro methods which were booming during this period. But despite 40 years of heavy investments, industrial H. brasiliensis clones could still not be mass micropro- pagated in vitro efficiently enough to meet the requirements of large scale produc- tion. The situation may change radically soon, however, due to the development of new nursery techniques adapted to the mass clonal production by rooted cuttings of any H. brasiliensis selected genotype. Efforts to improve the techniques as well as the establishment of new field trials are underway in order to determine if self-rooted rubber tree clones are more productive than grafted ones. This old issue is becoming of overriding impor- tance considering the increasing pres- sure on land availability reducing thereby the prospects for expanding rubber tree plantations.

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50

Zhu, Bao-Xuan, and Qingxiu Wang. "Unimodality of independence polynomials of rooted products of graphs." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 5 (June 4, 2019): 2573–85. http://dx.doi.org/10.1017/prm.2019.23.

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AbstractIn 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.

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