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On the dimension of trees

S. Poljak, A. Pultr
1981 Discrete Mathematics  
It is prov& that for trees (and forests) G one has dim G ~3 log; IG\ (where dim G, the dimension ci G, is the minimum k such that G is embeddable into a pfoduct of k complete graphs; ICI is the size of  ...  Moreover, if m(G) is the quotient ti' G obtained by identifying the points with &nciding neighbour sets, one has, for foksts G, log,' Im(G)f -1 adim G ~3 log; r&G)+ 1.  ...  According to 1 S, In general, we have alrbitrarily dimension 2). the assumption of m(G) = G in 2.5 is essential: large trees of dimension 2 (e.g., all stars have 3.  ... 
doi:10.1016/0012-365x(81)90064-9 fatcat:aakxkvs3h5ghjpk6rh7bkgsn6y

On the partition dimension of trees

Juan A. Rodríguez-Velázquez, Ismael González Yero, Magdalena Lemańska
2014 Discrete Applied Mathematics  
The partition dimension of G is the minimum number of sets in any resolving partition of G. In this paper we obtain several tight bounds on the partition dimension of trees.  ...  ,P_t} of the vertex set V of a connected graph G=(V,E), the partition representation of a vertex v∈ V with respect to the partition Π is the vector r(v|Π)=(d(v,P_1),d(v,P_2),...  ...  On the partition dimension of generalized trees A cut vertex in a graph is a vertex whose removal increases the number of components of the graph and an extreme vertex is a vertex such that its closed  ... 
doi:10.1016/j.dam.2013.09.026 fatcat:4webglg22neyfaouodyiiuhgpy

On the decomposition dimension of trees

Hikoe Enomoto, Tomoki Nakamigawa
2002 Discrete Mathematics  
The relation between the decomposition dimension and the diameter of a tree is also discussed.  ...  The decomposition dimension dec(G) of a graph G is the least integer r such that there exists a resolving r-decomposition.  ...  In this section, we consider the relation between the decomposition dimension and the diameter of a tree. Let T be a tree and let x be a vertex of T .  ... 
doi:10.1016/s0012-365x(01)00454-x fatcat:zzvwlgvkjfcilianpkovwhawiq

Hausdorff dimension of some groups acting on the binary tree

Olivier Siegenthaler
2008 Journal of group theroy  
Based on the work of Abercrombie, Barnea and Shalev gave an explicit formula for the Hausdorff dimension of a group acting on a rooted tree. We focus here on the binary tree T.  ...  In this article we explicitly compute the Hausdorff dimension of the level-transitive spinal groups.  ...  This is achieved by computing the dimension of the so-called spinal groups acting on the binary tree, which are generalizations of the Grigorchuk groups.  ... 
doi:10.1515/jgt.2008.034 fatcat:qeinricvnvhszkvm5qpzeoaxu4

On the spectral dimension of random trees

Bergfinnur Durhuus, Thordur Jonsson, John Wheater
2006 Discrete Mathematics & Theoretical Computer Science  
International audience We determine the spectral dimensions of a variety of ensembles of infinite trees.  ...  Among these ensembles is the so-called uniform random tree. We show that generic random trees have spectral dimension $d_s=4/3$.  ...  The spectral dimension is one example of such a quantity.  ... 
doi:10.46298/dmtcs.3507 fatcat:wrikpddvovhrjheonpap7hqjzq

On the SIG dimension of trees under L_∞ metric [article]

L. Sunil Chandran, Rajesh Chitnis, Ramanjit Kumar
2011 arXiv   pre-print
We study the SIG dimension of trees under L_∞ metric and answer an open problem posed by Michael and Quint (Discrete Applied Mathematics: 127, pages 447-460, 2003).  ...  Let T be a tree with atleast two vertices. For each v∈ V(T), let leaf-degree(v) denote the number of neighbours of v that are leaves.  ...  Hence t ≥ ⌈log(β + 1)⌉. ⊓ ⊔ 6 Upper Bound for SIG dimension of trees under L ∞ metric 6.1 Basic Notation For a non-leaf vertex x of the rooted tree T ′ , let C(x) denote the children of x.  ... 
arXiv:0910.5380v2 fatcat:x6flgbz6ejf3rl4krejhxjetam

A Note on the Dimensions of the Bronchial Tree

C. H. Barnett
1957 Thorax  
The theoretical implications of their observations will be discussed. MATERIAL AND METHODS For measuring the cross-sectional area of the bronchial tree, one of a series of resin casts prepared by Dr.  ...  smaller one diverges most from the direction of the parent tube."  ... 
doi:10.1136/thx.12.2.175 pmid:13442963 pmcid:PMC1019245 fatcat:diammw2jinesxeyctz7tno24ma

Bounding the Order of a Graph Using Its Diameter and Metric Dimension: A Study Through Tree Decompositions and VC Dimension

Laurent Beaudou, Peter Dankelmann, Florent Foucaud, Michael A. Henning, Arnaud Mary, Aline Parreau
2018 SIAM Journal on Discrete Mathematics  
The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set.  ...  We prove a bound of the form n=O(kd^2) for trees and outerplanar graphs (for trees we determine the best possible bound and the corresponding extremal examples).  ...  The authors are grateful to Julien Cassaigne, who found the family of planar graphs described in the conclusion, and to the anonymous referees for their careful reading and constructive remarks.  ... 
doi:10.1137/16m1097833 fatcat:sqtcd5u2nvf3fnzjandj7umbza

Local dimensions of the branching measure on a Galton–Watson tree

Q Liu
2001 Annales de l'I.H.P. Probabilites et statistiques  
Let µ = µ ω be the branching measure on the boundary ∂T of a supercritical Galton-Watson tree T = T(ω). Denote by d (µ, u) and d (µ, u) the lower and upper local dimensions of µ at u ∈ ∂T.  ...  Here we find exactly when the result holds for all u ∈ ∂T, and obtain some limit theorems about the uniform local dimensions of µ.  ...  Introduction We always assume that p 0 = 0, that N = N ∅ is not almost surely (a.s.) constant, and that For all u ∈ U, let T u be the shifted tree of T at u: this is the tree with defining elements {N  ... 
doi:10.1016/s0246-0203(00)01065-7 fatcat:bf3o6swvybax7huc4jw6d4sapq

The Dependence of Volume Increment of Individual Trees on Dominance, Crown Dimensions, and Competition

G. J. HAMILTON
1969 Forestry (London)  
The Dependence of Volume Increment of Individual Trees on Dominance, Crown Dimensions, and Competition G. J.  ...  On the face of it, it would seem that g.b.h. is basically an expression of dominance, and given similar crown dimensions it is reasonable to expect a tree occupying a higher position in the canopy would  ... 
doi:10.1093/forestry/42.2.133 fatcat:sojnvvhpk5d2hooet6cfhyc3wy

A note on the cubical dimension of new classes of binary trees

Kamal Kabyl, Abdelhafid Berrachedi, Éric Sopena
2015 Czechoslovak Mathematical Journal  
The conjecture of Havel [On hamiltonian circuits and spanning trees of hypercubes. CasopisP est.  ...  The cubical dimension of a graph G is the smallest dimension of a hypercube into which G is embeddable as a subgraph.  ...  The hypercube of dimension n, denoted Q n , is the graph whose 2 n vertices are boolean vectors of length n, such that two vertices are adjacent if and only if they differ in exactly one coordinate.  ... 
doi:10.1007/s10587-015-0165-6 fatcat:4zdrilk6efcwjeult5b6gv6rga

Calculating the VC-dimension of decision trees

Ozlem Asian, Olcay Taner Yildiz, Ethem Alpaydin
2009 2009 24th International Symposium on Computer and Information Sciences  
The VC-dimension of the univariate decision tree with binary features depends on (i) the VC-dimension values of the left and right subtrees, (ii) the number of inputs, and (iii) the number of nodes in  ...  From a training set of example trees whose VC-dimensions are calculated by exhaustive search, we fit a general regressor to estimate the VC-dimension of any binary tree.  ...  Due to the exponential time complexity of the algorithm, it is applied up to five dimensional inputs and on decision trees with depth up to five.  ... 
doi:10.1109/iscis.2009.5291847 dblp:conf/iscis/AsianYA09 fatcat:zsmotpm3azh77nbr7z4a5q7x5a

Notes on the Relative Age and Dimensions of a Number of Different Trees

N. L. Britton
1879 Bulletin of the Torrey Botanical Club  
of the number of years wlich must elapse before we may expect a sapling to become a tree of given dimensions.  ...  The subject of the annual growtlh of trees is one about which little seems to be known in comparison with its importan-ce.  ... 
doi:10.2307/2475632 fatcat:pbq5q4mzjbaz3d32bhwm4kkx7i

Note On A Scots Pine Tree Of Great Dimensions In Co. York

Dyce Duckworth
1908 Transactions of the Botanical Society of Edinburgh  
LL.D. (1908) Note On A Scots Pine Tree Of Great Dimensions In Co.  ...  NOTE ON A SCOTS PINE TREE. 321 NOTE ON A SCOTS PINE TREE OF GREAT DIMENSIONS IN CO. CORK. 1 By Sir DYCE DUCKWORTH, M.D., LL.D. Mr.  ... 
doi:10.1080/03746600809469174 fatcat:htotieegyvebhetq3hcyn7qq74

An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree

Steven P. Lalley, Thomas Sellke
2000 Probability theory and related fields  
Then the Hausdorff dimension of ∩ −1 ( µ ) in the metric d is almost surely on the event of nonextinction, where h(µ) is the entropy of the measure µ and q(i, j ) is the mean number of type-j offspring  ...  Let T be the genealogical tree of a supercritical multitype Galton-Watson process, and let be the limit set of T, i.e., the set of all infinite self-avoiding paths (called ends) through T that begin at  ...  If the offspring distribution has mean α > 1 and finite second moment then, almost surely on the event of nonextinction, the limit set of the Galton-Watson tree has Hausdorff dimension (in the metric d  ... 
doi:10.1007/pl00008722 fatcat:t5ue3xv7nrhdzg2nvoxx2odtve
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