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On the dimension of trees

1981
*
Discrete Mathematics
*

It is prov& that for

doi:10.1016/0012-365x(81)90064-9
fatcat:aakxkvs3h5ghjpk6rh7bkgsn6y
*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. ...##
###
On the partition dimension of trees

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 ...

##
###
On the decomposition dimension of trees

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 . ...

##
###
Hausdorff dimension of some groups acting on the binary tree

2008
*
Journal of group theroy
*

Based

doi:10.1515/jgt.2008.034
fatcat:qeinricvnvhszkvm5qpzeoaxu4
*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. ...##
###
On the spectral dimension of random trees

2006
*
Discrete Mathematics & Theoretical Computer Science
*

International audience We determine

doi:10.46298/dmtcs.3507
fatcat:wrikpddvovhrjheonpap7hqjzq
*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. ...##
###
On the SIG dimension of trees under L_∞ metric
[article]

2011
*
arXiv
*
pre-print

We study

arXiv:0910.5380v2
fatcat:x6flgbz6ejf3rl4krejhxjetam
*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. ...##
###
A Note on the Dimensions of the Bronchial Tree

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." ...

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

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. ...

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

2001
*
Annales de l'I.H.P. Probabilites et statistiques
*

Let µ = µ ω be

doi:10.1016/s0246-0203(00)01065-7
fatcat:bf3o6swvybax7huc4jw6d4sapq
*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 ...##
###
The Dependence of Volume Increment of Individual Trees on Dominance, Crown Dimensions, and Competition

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 ...

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

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. ...

##
###
Calculating the VC-dimension of decision trees

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. ...

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

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. ...

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

1908
*
Transactions of the Botanical Society of Edinburgh
*

LL.D. (1908) Note

doi:10.1080/03746600809469174
fatcat:htotieegyvebhetq3hcyn7qq74
*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. ...##
###
An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree

2000
*
Probability theory and related fields
*

Then

doi:10.1007/pl00008722
fatcat:t5ue3xv7nrhdzg2nvoxx2odtve
*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 ...
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