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A First Investigation of Sturmian Trees
[chapter]

*
STACS 2007
*

We consider

doi:10.1007/978-3-540-70918-3_7
dblp:conf/stacs/BerstelBCF07
fatcat:ougbsfjqvnbirpbj6ocd4r57du
*Sturmian**trees*as*a*natural generalization*of**Sturmian*words.*A**Sturmian**tree*is*a**tree*having n+1 distinct subtrees*of*height n for each n. ... We give various examples*of**Sturmian**trees*, and we characterize one family*of**Sturmian**trees*by means*of**a*structural property*of*their automata. ... The*first*four words ε, 01, 0101 and 0011*of*L correspond to the four occurrences*of*the letter*a*as label on the top*of*the*tree*. ...##
###
Circular sturmian words and Hopcroft's algorithm

2009
*
Theoretical Computer Science
*

We prove that the shape

doi:10.1016/j.tcs.2009.07.018
fatcat:7x3khicglrcjnl3isvyh3avmvq
*of*the reduction*tree**of**a*circular*sturmian*word (x) coincides with that*of*the derivation*tree*T (*A*x )*of*the automaton*A*x . ... In order to analyze some extremal cases*of*Hopcroft's algorithm, we*investigate*the relationships between the combinatorial properties*of**a*circular*sturmian*word (x) and the run*of*the algorithm on the ... In the*first*part*of*the paper we*investigate*some combinatorial properties*of*circular*sturmian*words. ...##
###
Page 4406 of Mathematical Reviews Vol. , Issue 2002F
[page]

2002
*
Mathematical Reviews
*

*First*, we

*investigate*structural properties

*of*the executions

*of*such

*a*dynamic process graph Y. ... Sci. 255 (2001), no. 1-2, 539-553; MR 2002c:68073] begun

*a*study

*of*

*a*slightly more general class

*of*infinite words that we call epi-

*Sturmian*. ...

##
###
Substitution invariant Sturmian words and binary trees
[article]

2017
*
arXiv
*
pre-print

We show that homogeneous substitution invariant

arXiv:1705.08607v1
fatcat:uz3jp7m4jrgbhcpigv7lxk6fny
*Sturmian*sequences s_α,α can be indexed by two binary*trees*, associated directly to Johannes Kepler's*tree**of*harmonic fractions from 1619. ... We take*a*global view at substitution invariant*Sturmian*sequences. ... Below we display the*first*three levels*of*the*tree**of*Sturm numbers α with α < 1 2 associated to the morphisms*of*T ϕ . ...##
###
Infinite labeled trees: From rational to Sturmian trees

2010
*
Theoretical Computer Science
*

The

doi:10.1016/j.tcs.2009.12.009
fatcat:22hgfcxbpngn5mttfh7pstokhq
*first*part*of*the paper is dedicated to the study*of*general unordered infinite*trees*with binary labels. ... Section 3 is dedicated to the study*of*the rational*trees*. The second part*of*the paper*investigates*balanced unordered*trees*and their properties. ... Density*of**a*balanced*tree*Before beginning the full*investigation**of*balanced*trees*, we start with*a*rather straightforward property:*a*balanced*tree*has*a*density. ...##
###
Distributing Labels on Infinite Trees
[article]

2008
*
arXiv
*
pre-print

In this paper, we consider the problem

arXiv:0809.1989v1
fatcat:qj4mauinjba4zn6hikezdrwb4y
*of*generalizing*Sturmian*words to*trees*. The problem is to evenly distribute labels 0 and 1 over infinite*trees*. ... Therefore they bring the hope that extremal scheduling properties*of**Sturmian*words can be extended to such*trees*, as least partially. Such possible extensions are illustrated by one such example. ...*of*the paper*investigates*balanced unordered*trees*and their properties. ...##
###
Page 5108 of Mathematical Reviews Vol. , Issue 2000g
[page]

2000
*
Mathematical Reviews
*

Our main result shows that any spanning

*tree*defined by the edges corresponding to*a**first*visit*of**a*vertex 68 COMPUTER SCIENCE 5108 appears with*a*probability proportional to its weight, which is the ... the edges, are*investigated*. ...##
###
Lyndon factorization of sturmian words

2000
*
Discrete Mathematics
*

We also give

doi:10.1016/s0012-365x(99)00123-5
fatcat:rumv6a44l5ehrghysv4taw7wya
*a*short proof*of**a*result by Berstel and de Luca (*Sturmian*words, Lyndon words and*trees*, Theoret. Comput. ... We express any general characteristic*sturmian*word as*a*unique inÿnite non-increasing product*of*Lyndon words. Using this identity, we give*a*new !-division for characteristic*sturmian*words. ...*First*, we prove that the factorization*of*s gives an !-division for it. ...##
###
Page 8132 of Mathematical Reviews Vol. , Issue 2000k
[page]

2000
*
Mathematical Reviews
*

The performance

*of*the estimates is measured by the robust L, error. The convergence and rates*of*the estimates are*investigated*.” ... In the case*of*infinite words they prove that knowledge*of*the cardinality*of*M, is enough to characterize the set*of*factors*of*x in case x is*a**Sturmian*word (i.e.*a*word which has exactly n+1 different ...##
###
Page 3884 of Mathematical Reviews Vol. , Issue 98F
[page]

1998
*
Mathematical Reviews
*

Kapoor and Ramesh

*first*proposed an O(N + V + E)-time algorithm by adopting such*a*‘compact’ out- put, where N is the number*of*spanning*trees*. ... Shioura and*A*. Tamura [J. Oper. Res. Soc. Japan 38 (1995), no. 3, 331-344; MR 96f:05094]. They find*a*new spanning*tree*by exchanging one edge*of**a*cur- rent one. ...##
###
Quasi-Sturmian colorings on regular trees
[article]

2018
*
arXiv
*
pre-print

Quasi-

arXiv:1808.05400v1
fatcat:l66zwermhfhtzgpf3he27zxvrm
*Sturmian*words, which are infinite words with factor complexity eventually n+c share many properties with*Sturmian*words. In this paper, we study the quasi-*Sturmian*colorings on regular*trees*. ... There are two different types, bounded and unbounded,*of*quasi-*Sturmian*colorings. We obtain an induction algorithm similar to*Sturmian*colorings. We distinguish them by the recurrence function. ... Factor complexity and*Sturmian*words have been generalized to*Sturmian*colorings*of**trees*by the*first*and third authors [6] . ...##
###
Page 7292 of Mathematical Reviews Vol. , Issue 98K
[page]

1998
*
Mathematical Reviews
*

98k:68139
edge problem with respect to

*a*minimum spanning*tree*is to find*a*set S (C E)*of*k edges in G whose removal results in the greatest increase in the weight*of*the minimum spanning*tree*in the ...*of*their factors, or with the well-known*Sturmian*words). ...##
###
Page 6292 of Mathematical Reviews Vol. , Issue 2003h
[page]

2003
*
Mathematical Reviews
*

*A*

*Sturmian*sequence is an infinite binary word which contains exactly n +1 different factors

*of*length » for each nonnegative integer ».

*A*

*Sturmian*word is any finite factor

*of*

*a*

*Sturmian*sequence. ... The authors prove that if U is an unbordered

*Sturmian*word then every Duval extension W

*of*U is

*of*the form W = U"’ with ui . . :

*a*$ r = —~. ...

##
###
Bi-Lipshitz Embedding of Ultrametric Cantor Sets into Euclidean Spaces
[article]

2013
*
arXiv
*
pre-print

An ultrametric Cantor set can be seen as the boundary

arXiv:1202.4330v2
fatcat:r7nclmpnjfa45dih24mbwlzojy
*of**a*rooted weighted*tree*called the Michon*tree*. The notion*of*Assouad dimension is re-interpreted as seen on the Michon*tree*. ... It is applied to answer the embedding question for some spaces which can be seen naturally as boundary*of**trees*: linearly repetitive subshifts,*Sturmian*subshifts, and the boundary*of*Galton--Watson*trees*... B. thanks*A*. Grigor'yan and his group at the CRC701, University*of*Bielefeld for giving him the opportunity to present this result prior to publication.*A*. J. thanks J. ...##
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Rigidity and Substitutive Dendric Words

2018
*
International Journal of Foundations of Computer Science
*

We

doi:10.1142/s0129054118420017
fatcat:ljboyxp46ndfhgfvq265kt6sr4
*investigate*here the properties*of*substitutive*tree*words and prove some rigidity properties, that is, algebraic properties on the set*of*substitutions that fix*a**tree*word. ... This class*of*words includes classical families*of*words such as*Sturmian*and episturmian words, codings*of*interval exchanges or else Arnoux-Rauzy words. ... We*investigate*here the properties*of*substitutive*tree*words and prove some rigidity properties. ...
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