Questions tagged [trees]
A tree is a connected graph without cycles, with a finite or infinite number of vertices. There are many variants of trees, according to further constraints or decorations.
259 questions
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Two questions on forests with distinguished children
Let
$$X_0 \mathop \rightleftharpoons^{s_0}_{r_0} X_1 \mathop \rightleftharpoons^{s_1}_{r_1} X_2 \mathop \rightleftharpoons^{s_2}_{r_2} \cdots$$
be a diagram (in some bicomplete category) such that $...
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Spectraly extremal connected trees, a conjecture on majorisation of Laplacian eigenvalues
The precise conjecture that I want to prove or disprove is the following:
Let $P_n$ denote the path graph with n vertices, $S_n$ the star graph with n vertices, and $T_n$ an arbitrary connected tree ...
12
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Reference request for a proof of Cayley's tree counting formula via the representation theory of the symmetric group
Inspired by a recent project Euler problem, I came up with a proof (sketched below) of Cayley's tree formula using the representation theory of $S_n$. I would like to ask for a reference in the ...
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literature request: morphisms of plane trees as natural transformations in $[\omega^{op},\Delta_+]$?
This is a more open-ended followup to a related question: On Joyal's definition of a category of plane trees.
That question recalled how rooted plane trees can be represented as contravariant ...
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On Joyal's definition of a category of plane trees
I'm trying to understand better the motivation for the definition of the category $Trees$ of finite plane trees in Joyal's unpublished manuscript "Disks, duality and Θ-categories" (1997, ...
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References for incompleteness proofs using infinite trees or König's lemma
My professor mentioned that there is a proof of Gödel's first incompleteness theorem that uses combinatorial ideas such as König's lemma and the existence of infinite trees with finite branches, ...
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Does a random rooted tree with sufficiently many leaves almost surely contain a specific rooted tree as a subtree?
Let $\mathcal{T}_n$ be the set of rooted, unlabeled trees with $n$ leaves, where each vertex either has no child or has at least two children. Let $\mathcal{T} = \bigcup_{n \ge 2} \mathcal{T}_n$.
For ...
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Looking for: Neuer Beweis eines Satzes über Permutationen
Where do I find old publications like "Heinz Prüfer. 1918. Neuer Beweis eines Satzes über Permutationen. Archiv der Mathematik und Physik, 27:142–144."?
I was looking through librairies and ...
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2-splitting subtree of a 4-splitting tree with terminal nodes in a given set
Assume that we have a finite 4-splitting tree $T$ (that is, every non-terminal node has exactly 4 immediate successors) of height $h$, and that we choose an arbitrary subset $A$ of terminal nodes of $...
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Generate all rooted trees with a given number of vertices
A graph $G$ is said to be a tree if it is a connected (i.e. every two distinct vertices of $G$ can be connected by a path of vertices of $G$), acyclic (i.e., $G$ has no cycles) and simple (i.e., $G$ ...
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Conjectured graph transformations related to treewidth
This came when examining edge labeled graphs and is based on limited experimental evidence so counterexamples are likely.
Let $T(G)$ denote the treewidth of a graph.
Transformation (1) Set $G'=G$ and ...
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How is the graphical form of the subdivision of the multiplication and comultiplication defined on planar binary rooted trees? [closed]
How is the graphical form of the subdivision of the multiplication and comultiplication defined on planar binary rooted trees implemented in the paper? The paper is "Structure of the Loday–Ronco ...
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Tree with $1+\sqrt{2}$ an eigenvalue of its adjacency matrix
What is an example of a tree with $1+\sqrt{2}$ an eigenvalue of its adjacency matrix?
Such a tree must exist since "Every totally real algebraic integer is a tree eigenvalue", Justin Salez,...
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What is known about probability distributions on trees biased toward specific clades?
I am interested in a particular type of probability distribution over trees - given $n$ labelled leafs, I'd like to find out what kind of work has been done to characterize probability measures or ...
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Characterisations of trees
Here is a "big list" question.
What are the definitions of tree?
Here is what I have found.
Suppose that $\Gamma$ is a non-empty, connected graph.
[Graph-theory]
$\Gamma$ has no embedded ...
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Ramsey Theorem For $\omega$-branching trees
I've see a number of papers addressing both the computability theory and combinatorics of Ramsey's theory on binary trees. The following feels like a somewhat similar question on $\omega^{< \omega}$...
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Forest of growing weights in a graph
Given an undirected weighted graph $G = (V,E,\omega)$, we consider the relation $p[]$ defined by: for any vertex $v$, $p[v]$ is the vertex $u$ such that $\omega(v,u) = \max_{w\in N(v)} \omega(v,w)$ (...
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Matrix-tree theorem for inverse matrices
Let $L$ be the Laplacian of a directed weighted graph on $n$ nodes, e.g., for $n=4$:
$$
L = \left(\begin{array}{cccc} w_{1,1}+w_{1,2}+w_{1,3}+w_{1,4} & -w_{1,2} & -w_{1,3} & -w_{1,4}\\ ...
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Determinant formula for a certain parametrized M-matrix
Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by
$$
A_{ij} = \begin{cases}
-P_{ij} & i \neq j,\\
P_{i1} + P_{i2} + \dots + P_{in} & i=j.
\end{cases}
$$...
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Name For Effective Cantor-Bendixsonish Derivitive
When dealing with a tree (substring closed subset of $\omega^{< \omega})$ a useful operation will frequently be to remove any nodes with finite ordinal rank (i.e., all nodes whose extensions on the ...
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Is the poset in the following construction stationary $\aleph_{\alpha + 2}$-linked?
Definition: A poset P is $\mathbf{stationary}$ $\kappa^{+}$-$\mathbf{linked}$ if for every sequence of conditions $(p_{\gamma} | \gamma < \kappa^{+})$, there is a regerssive function $f: \kappa^{+} ...
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Counting with trees
Let $\mathcal{U}_n$ denote the set of unrooted unlabelled trees with $n$ edges. For $T\in\mathcal{U}_n$, let $1^{u_1}2^{u_2}\cdots n^{u_n}$ be its degree distribution, that is, $u_i=\#$ of vertices ...
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4
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Properties of all relatively computable branches
I'm probably just missing something obvious but suppose that $T \subset 2^{< \omega}$ is a perfect tree with no terminal nodes (what about just $[T]$ non-empty?). If $Y \leq_{T} X$ for all $X \...
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Finding the diameter of an unknown tree: Is BFS optimal?
I'm interested on the following nice problem that is somewhat standard in CS, but I was surprised on the lack of references on the optimal algorithm to this problem.
Ana and Banana plays the ...
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2
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338
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Multiplicity of the smallest non-zero Laplacian eigenvalue for tree graphs
How accurate is the following statement: "For tree graphs, the multiplicity of the smallest non-zero eigenvalue $\lambda_2$ of the Laplacian is 1." If not valid, in which cases does it fail ...
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Is it consistent to have $\kappa$-Kurepa trees for some $\kappa$, but no Kurepa trees of other heights?
Definitions A tree means a set-theoretic tree, that is a poset $(T,<)$ so that for each $x\in T$, the set $\{y\in T\mid y<x\}$ is well-ordered.
A $\kappa$-Kurepa tree is a tree of height $\kappa$...
1
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Two independent spanning trees of $2$-connected graph with $P_5$-free and $K_{1,3}$-free
I'm going to prove the following statement:
$G$ is a $P_5$-free and $K_{1,3}$-free graph with $\vert G \vert \geq 7$, and $G \notin \mathcal{K}$, then $G$ is $2$-connected graph if and only if $G$ ...
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Name of this type of graph?
What is the name of a graph that has $1$ central node connected to $n$ other nodes, each of them connected to $n-1$ distinct nodes, and so on?
At the end of the process the central node has degree $n$,...
1
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0
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98
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Set-theoretic trees with ordering between siblings
In set-theoretic trees, we have the binary relation $<_T$ (which defines the "ancestor-descendant" ordering). This relation is partial, as children are incomparable in that ordering.
...
4
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1
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Simplified method of building an Aronszajn tree
There is a very interesting method to build an Aronszajn tree in Judith Roitman's "Introduction to Modern Set Theory", on pages 100-102. In short, we build a tree $T$ in which the nodes are ...
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Name for generalization of trees to digraphs
One definition of tree in graph theory could be as follows:
A tree is a(n undirected) graph for which there is a unique (undirected) path between any pair of vertices.
This suggest a possible ...
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Existence of trees with height $\kappa$, every level has at most size $\lambda$ and has at least $\lambda^{+}$ maximal branches
Definitions A tree means a set-theoretic tree, that is a poset $(T,<)$ so that for each $x\in T$, the set $\{y\in T\mid y<x\}$ is well-ordered.
A $\kappa$-Kurepa tree is a tree of height $\kappa$...
4
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Is the integer factorization into prime numbers normally distributed?
Edit:
Sorry, for the inconvenience: I have edited the question, since there was a misconception in my thinking.
Let $P_1(n) := 1$ if $n=1$ and $\max_{q\mid n, \text{ } q\text{ prime}} q$ otherwise, ...
0
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2
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432
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Factorization trees and (continued) fractions?
This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question:
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
4
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2
answers
558
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Existence of trees with height $\omega$, size $\aleph_1$ and $\aleph_2$ maximal branches
Definition A tree means a set-theoretic tree, that is a poset $(T,<)$ so that for each $x\in T$, the set $\{y\in T\mid y<x\}$ is well-ordered.
Question: I would like to know if it is consistent ...
0
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Approximating all spanning trees with their sample
In a complete graph with $n$ vertices there are $n^{n-2}$ trees.
In my research I'm analyzing trees in the following way (each edge has a weight):
Get a tree.
Build a complete graph, by the following ...
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Strong law of large numbers for a sequence of random variables in different probability spaces
Is it known whether the following version of the strong law of large numbers holds?
For each $k\in\mathbb{N}$, let $\Omega_k$ be a finite set and $\mu_k$ be a probability measure on $\Omega_k$. Let $(...
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Thick Canadian trees
A Canadian tree (also called a weak Kurepa tree) is a tree of height $\omega_1$ with levels of cardinality $\leq \omega_1$ and at least $\omega_2$ many uncountable branches. Let's call a Canadian tree ...
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Action of braid groups on regular trees
Question:
Are there any well known actions of braid groups on trees? For example is there some action of a braid group $ B_n $ on a $ p $ regular tree for some $ p $ such that the action is transitive ...
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Fractal dimension of a self-similar tree
Consider a binary tree constructed as the following. Given a node with a some value $x$, I construct two children nodes each having value $l(x)$ and $r(x)$ respectively. I repeat the same on the ...
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Maximal cliques in neighborhood graphs of partial $k$-trees (bounded treewidth)
Background
My question is about a generalization of the following situation:
Let $M$ be a finite metric space. Given $r>0$, the $r$-neighborhood graph $N(M)_r$ has vertex set $M$ and an edge $\{x,y\...
0
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1
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85
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Minimum spanning tree and projection
Let $G$ be a graph of $n$ arcs and let $x\in \mathbb{R}^n$. I want to compute the orthogonal projection of $x$ onto the set of radial graphs with $k$ roots contained in $G$ (or a forest with $k$ root) ...
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Terminology for a subtree of a rooted tree with a path boundedness property
I'm not a graph theorist, so I apologize if some of the following terminology isn't quite correct.
Let $(T,f,v_0)$ be a complete degree $d$ rooted tree (definition at the end).
Definition. Let $m\ge0$....
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Characterization of graphs without leaves
Let $G(n,l)$ denote the set of connected graphs with $n$ vertices and $l$ edges and let $G_0(n,l)$ denote the elements of $G(n,l)$ without leaves. It is easily seen that $G_0(n,n-1)$ is empty, since $...
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Structure Theory for Tree Decompositions
I that $G=(V,E,W)$ is a weighted graph with positive edge weights and a finite set of vertices $K$. Let $0\le k,M\le K$ be a fixed integer.
Is is known when $G$ admits the following type of ...
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Standard terminology for node in tree with multiple children
Is there a standard terminology for a node in a tree that has multiple children?
For instance, in describing in perfect tree in $\omega^{< \omega}$ how would you describe the nodes that are ...
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A combinatorial interpretation for $n$-ary trees for negative $n$
The ordinary generating function $T_n=T_n(x)$ for the $n$-ary trees satisfies the functional equation
$$
T_n=1+xT_n^n.
$$
This is usually defined for $n\ge 0$, but the functional equation can be ...
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Cofinal trees in $({}^\omega \omega , \leq^\ast )$
So, I know that the existence of a scale (that is, a linear cofinal set in $({}^\omega \omega , \leq^\ast )$, where $\leq^\ast $ is eventual domination, is equivalent to $\mathfrak{b} = \mathfrak{d}$, ...
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What is the analogue of a Block-Cut Tree Decomposition in directed graphs?
Let $G$ be a connected, undirected graph. We define a block $B$ to be a maximal $2$-connected induced subgraph in $G$. It is easy to see that any two distinct blocks are either disjoint or overlap at ...
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In search of a quick proof that groups acting freely on $\mathbb R$-trees are linear
Many years ago I had the idea to use non-standard analysis to prove that a group acting freely on an $\mathbb R$-tree must be linear. The heuristic went like this:
A non-standard model $G^*$
of the ...
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