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### Counting nodes in binary trees

Sami Khuri
<span title="">1986</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/42mrbbmjbfdtbbepad2fzixmum" style="color: black;">Proceedings of the seventeenth SIGCSE technical symposium on Computer science education - SIGCSE &#39;86</a> </i> &nbsp;
One can get the closed form of T(n) by counting the number of internal nodes of the trimmed tree corresponding to n. The count is done level by level.  ...  Examples -- -- 1) n=4 ; x1x2x3x4+ x1x2x3x4+ x1x2x3x4+ ---- x1x2x3X4 Note that in figure 2, X . is used at level i for i = 1,2,3,4 in the trimmed tree. X.. the trees in figures 2 and 3.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/5600.5646">doi:10.1145/5600.5646</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/sigcse/Khuri86.html">dblp:conf/sigcse/Khuri86</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zbj5mf65azahnjte6ca4qmz44q">fatcat:zbj5mf65azahnjte6ca4qmz44q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20101217031242/http://www.cs.sjsu.edu/%7Ekhuri/cse86.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f2/d0/f2d04c446daebade9baf626a3f71a3b3de421527.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/5600.5646"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

### Counting in Trees for Free [chapter]

Helmut Seidl, Thomas Schwentick, Anca Muscholl, Peter Habermehl
<span title="">2004</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Further, to decide whether a tree Ø satisfies a formula ³ is polynomial in the size of ³ and linear in the size of Ø.  ...  It is known that MSO logic for ordered unranked trees is undecidable if Presburger constraints are allowed at children of nodes.  ...  Since the class of tree languages defined by deterministic PTA is a strict superclass of the regular tree languages, we would also like to see other characterizations of this class.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-27836-8_94">doi:10.1007/978-3-540-27836-8_94</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xa5ds54akbad5h6ex2mtmqm5x4">fatcat:xa5ds54akbad5h6ex2mtmqm5x4</a> </span>
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### Counting nodes in binary trees

Sami Khuri
<span title="1986-02-01">1986</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qzoq6upx4ndyrd7qaolzbaeuca" style="color: black;">ACM SIGCSE Bulletin</a> </i> &nbsp;
One can get the closed form of T(n) by counting the number of internal nodes of the trimmed tree corresponding to n. The count is done level by level.  ...  Examples -- -- 1) n=4 ; x1x2x3x4+ x1x2x3x4+ x1x2x3x4+ ---- x1x2x3X4 Note that in figure 2, X . is used at level i for i = 1,2,3,4 in the trimmed tree. X.. the trees in figures 2 and 3.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/953055.5646">doi:10.1145/953055.5646</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6tsogiuabvdplmjygerr7v47uu">fatcat:6tsogiuabvdplmjygerr7v47uu</a> </span>
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### Counting paths in perfect trees [article]

Peter J. Humphries
<span title="2017-11-23">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We first count the paths in perfect rooted m-ary trees and then use the results to determine the number of paths in perfect unrooted m-ary trees, extending a known result for binary trees.  ...  We present some exact expressions for the number of paths of a given length in a perfect m-ary tree.  ...  In this paper, we focus on counting the pairs of vertices that are some given distance apart, or equivalently the paths of a given length, in a perfect tree.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1711.08555v1">arXiv:1711.08555v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/53p3wcyccfg4hgw5anolfqs36u">fatcat:53p3wcyccfg4hgw5anolfqs36u</a> </span>
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### Counting Trees in Supersymmetric Quantum Mechanics [article]

Clay Cordova, Shu-Heng Shao
<span title="2015-03-02">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The ground state degeneracy may be written as a multi-dimensional contour integral, and the enumeration of poles can be simply phrased as counting bipartite trees.  ...  This is the simplest quiver with two gauge groups and bifundamental matter fields, and appears universally in four-dimensional N=2 systems.  ...  Elkies, Ira Gessel, Ying-Hsuan Lin, and Alexey Ustinov for many crucial steps in the proofs.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1502.08050v2">arXiv:1502.08050v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vn6eb4ffxngkdhpmx6ubunvpba">fatcat:vn6eb4ffxngkdhpmx6ubunvpba</a> </span>
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### Counting Branches in Trees Using Games [article]

Arnaud Carayol, Axel Haddad, Olivier Serre
<span title="2015-05-14">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In the case (ii) where one counts accepting branches it leads to new proofs (without appealing to logic) of an old result of Beauquier and Niwinski.  ...  We study finite automata running over infinite binary trees. A run of such an automaton is usually said to be accepting if all its branches are accepting.  ...  Moreover, in the case where one counts accepting branches we show that the languages that we obtain are always accepted by a Büchi automaton, which contrasts with the case where one counts rejecting branches  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1505.03852v1">arXiv:1505.03852v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/75ka65bvw5gxhpbavalbces55m">fatcat:75ka65bvw5gxhpbavalbces55m</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200916150231/https://arxiv.org/pdf/1505.03852v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ef/5f/ef5f753a71f39c69a4dfe5cf80d58253c6ff0a88.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1505.03852v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Counting trees in supersymmetric quantum mechanics

Clay Córdova, Shu-Heng Shao
<span title="2018-02-01">2018</span> <i title="European Mathematical Publishing House"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3ogj2khcpnccpmr6nnk27vziue" style="color: black;">Annales de l&#39;Institut Henri Poincaré D</a> </i> &nbsp;
The ground state degeneracy may be written as a multi-dimensional contour integral, and the enumeration of poles can be simply phrased as counting bipartite trees.  ...  This is the simplest quiver with two gauge groups and bifundamental matter fields, and appears universally in four-dimensional N = 2 systems.  ...  Elkies, Ira Gessel, Ying-Hsuan Lin, and Alexey Ustinov for many crucial steps in the proofs.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4171/aihpd/47">doi:10.4171/aihpd/47</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qyhiaayt5vdbfjgptmmew7ssze">fatcat:qyhiaayt5vdbfjgptmmew7ssze</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200322170514/http://inspirehep.net/record/1346868/files/arXiv:1502.08050.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/bf/4d/bf4dda58fe697c1a6f24b1479927439a0e821127.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4171/aihpd/47"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

### Counting lattices in products of trees [article]

Nir Lazarovich, Ivan Levcovitz, Alex Margolis
<span title="2022-02-01">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A BMW group of degree (m,n) is a group that acts simply transitively on vertices of the product of two regular trees of degrees m and n.  ...  In fact, we show that the same bounds hold for virtually simple BMW groups.  ...  Counting BMW groups. Let BMW(m, n) be the set of all BMW groups of degree (m, n) up to conjugacy in Aut(T m ) × Aut(T n ).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2202.00378v1">arXiv:2202.00378v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qcpezr6x5zg7behmgoouidk5qm">fatcat:qcpezr6x5zg7behmgoouidk5qm</a> </span>
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### Counting branches in trees using games

Arnaud Carayol, Olivier Serre
<span title="">2017</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/joe2ngto45hbnl3pncnesnq344" style="color: black;">Information and Computation</a> </i> &nbsp;
In the case (ii) where one counts accepting branches it leads to new proofs (without appealing to logic) of a result of Beauquier and Niwiński.  ...  We study finite automata running over infinite binary trees. A run of such an automaton is usually said to be accepting if all its branches are accepting.  ...  Moreover, in the case where one counts accepting branches we show that the languages that we obtain are always accepted by a Büchi automaton, which contrasts with the case where one counts rejecting branches  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ic.2016.11.005">doi:10.1016/j.ic.2016.11.005</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hk5mbamj5vblfkddok6fwfljje">fatcat:hk5mbamj5vblfkddok6fwfljje</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180727183420/https://hal.archives-ouvertes.fr/hal-01563189/document" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/86/98/869838990d0cc55fe0aeeaad31fb57f3b89f521e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ic.2016.11.005"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

### Faster algorithms for counting subgraphs in sparse graphs [article]

Marco Bressan
<span title="2020-08-30">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We answer in the affirmative by introducing a novel tree-like decomposition for directed acyclic graphs, inspired by the classic tree decomposition for undirected graphs.  ...  In this work we address the following question: can we count the copies of H faster if G is sparse?  ...  In this work we introduce a novel tree-like graph decomposition, to be applied to the pattern graph H, designed to exploit the degeneracy of G when counting the homomorphisms of H in G.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.02089v5">arXiv:1805.02089v5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/s6lih2ah4vbmblif3v3kfp4g5y">fatcat:s6lih2ah4vbmblif3v3kfp4g5y</a> </span>
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### Counting Spanning trees in double nested graphs [article]

Fernando Tura
<span title="2016-05-16">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper we give a linear time algorithm for computing the number of spanninig trees in double nested graphs.  ...  The Kirchhoff Matrix Tree Theorem is one of most important results in graph theory.  ...  The problem of computing the number of spanning trees on the graph G is an important problem in graph theory.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1605.04760v1">arXiv:1605.04760v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bdxwsx6wmbenjid5od3tbtsabi">fatcat:bdxwsx6wmbenjid5od3tbtsabi</a> </span>
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### Counting overlattices in automorphism groups of trees [article]

Seonhee Lim
<span title="2005-06-11">2005</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We give an upper bound for the number of "overlattices" in the automorphism group of a tree, containing a fixed lattice with index n.  ...  For an example of a lattice in the automorphism group of a 2p-regular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well.  ...  Counting overlattices Let Γ be a cocompact lattice in Aut(T ).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0506217v1">arXiv:math/0506217v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/a2tjhqvkujdihcia7fpvkjh5km">fatcat:a2tjhqvkujdihcia7fpvkjh5km</a> </span>
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### Orbit counting in conjugacy classes for free groups acting on trees [article]

George Kenison, Richard Sharp
<span title="2016-06-30">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper we study the action of the fundamental group of a finite metric graph on its universal covering tree.  ...  We obtain an asymptotic for the number of elements in a fixed conjugacy class for which the associated displacement of a given base vertex in the universal covering tree is at most T.  ...  In order to study the counting problem in the conjugacy class C, we define a generating function η C (s) = x∈C e −sL(x) .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1502.02591v3">arXiv:1502.02591v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qzbsqvzwofh45bfqd7xux3rhsm">fatcat:qzbsqvzwofh45bfqd7xux3rhsm</a> </span>
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### A Beginner's Guide to Counting Spanning Trees in a Graph [article]

Saad Quader
<span title="2012-08-01">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
(DRAFT VERSION) In this article we present a proof of the famous Kirchoff's Matrix-Tree theorem, which relates the number of spanning trees in a connected graph with the cofactors (and eigenvalues) of  ...  This is a 165 year old result in graph theory and the proof is conceptually simple.  ...  Conclusion In this paper, we have presented a proof of the Kirschhoff's matrix tree theorem which gives a formula of counting the number of spanning trees in a graph in terms of the eigenvalues of the  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1207.7033v2">arXiv:1207.7033v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oqootixv45gvvgwgfskgg5wzi4">fatcat:oqootixv45gvvgwgfskgg5wzi4</a> </span>
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### Counting contours on trees

Noga Alon, Rodrigo Bissacot, Eric Ossami Endo
<span title="2016-11-24">2016</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/viusnta3dbaqlbghkof3zr2ge4" style="color: black;">Letters in Mathematical Physics</a> </i> &nbsp;
We calculate the exact number of contours of size n containing a fixed vertex in d-ary trees and provide sharp estimates for this number for more general trees.  ...  We also obtain a characterization of the locally finite trees with infinitely many contours of the same size containing a fixed vertex.  ...  this note with colleagues at ICMC-USP and UFSCAR in São Carlos, Brazil.  ...
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