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.

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What is a universal tree?

I came across some slides talking about the Hrushovski construction. One of the examples was the construction of a "universal tree". I was curious because the collection of finite trees does not ...
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170 views

Finding many disjoint sub-trees with many leaves

Let $T$ be a rooted binary tree with $L$ leaves, and let $\ell$ be a natural number smaller than $L$. The question is what is the maximal number of disjoint rooted sub-trees with at least $\ell$ ...
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Solving Recursive Expression for Counting Unrooted Tree Topologies

The book Bayesian Evolutionary Analysis with BEAST by Alexei J. Drummond et al. (2015) states at section 2.2.1 that the number of rooted unlabelled (binary) tree topologies $a_n$ is given by the ...
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Translation length functions of non-simplicial trees

Let $G$ be a finitely generated group. By a theorem of Culler and Morgan, the set of non-abelian (not necessarily simplicial) minimal $\mathbb{R}$-trees with isometric $G$-action injects into the ...
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Is there an efficient algorithm to find all the maximum matching in any tree?

A matching in a graph (G) is a set of mutually non-adjacent edges of (G). A maximum matching is a matching of maximal cardinality. A tree is an acyclic connected graph. Is there an efficient ...
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Similarity metric for labelled weighted graph/minimum spanning tree

I'm looking for a metric to measure similarity of minimum spanning trees of labelled weighted graphs. Each entity to compare has the same nodes (number of nodes and labels identical) but (unlabelled) ...
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1answer
140 views

Can a surface group act on a finite-valence simplicial tree?

Question. Let $S$ be a closed surface of genus $> 1$. Can $\pi_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree. ...
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164 views

A published proof for: the number of labeled $i$-edge ($i \geq 1$) forests on $p^k$ vertices is divisible by $p^k$

Let $F(n;i)$ be the number of labeled $i$-edge forests on $n$ vertices (A138464 on the OEIS). The first few values of $F(n;i) \pmod n$ are listed below: $$\begin{array}{r|rrrrrrrrrrr} & i=0 &...
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Tree decompositions in linear hypergraphs

A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a finite set and $L\subseteq {\cal P}(X)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \in L$ ...
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71 views

Factors between IID on trees: what about the useless information?

Let $p \in (0,1)$. Take $E$ to be the edge set of the trivalent tree $T$, and $G$ to be the automorphism group of $T$. Let $f$ be any $G$-equivariant map from the measure space $([0,1]^E, \text{d}x^{\...
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321 views

To what extent is it possible to generalise a natural bijection between trees and $7$-tuples of trees, suggested by divergent series?

This is a cross-post from MSE. In the paper "Seven Trees In One" by Andreas Blass, a "very explicit" bijection is found between trees and 7-tuples of such trees. The idea to construct such a ...
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257 views

Weighted sum of the Simsun (Andre) permutations

Let $ c_{n,k} $ be the Simsun permutations$^1$ defined by the following relations: $\displaystyle c_{n,0} = 1, \hspace{0.1cm} (n \ge 1);$ $$ c_{n,k} = (k+1) c_{n-1,k} +(n-2k+1) c_{n-1,k-1}, \hspace{0....
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1answer
140 views

Counting growing tree trajectories

I am looking for help: Beginning with a single node ($\circ$), at each discrete time step I can add a node/link pair to any node currently in the tree. Nodes are unlabelled and the tree is ...
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143 views

Database of non-isomorphic trees

As there are several free prime number databases, is there something similar for non-isomorphic trees?
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280 views

Partitioning a binary tree into vertex-disjoint subtrees

Say we have a labeled, binary unrooted tree $T$, i.e. each node has either 1 or 3 neighbors. Denote by $L(T)$ the set of leaves (degree-one nodes) of $T$. For some $L \subseteq L(T)$, denote by $t(L)$...
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1answer
89 views

Spectrum of Laplacian matrix of an infinite tree graph

I'm having difficulty understanding a fact stated in a research paper I'm reading. Namely, let $T$ be a tree with all nodes of degree $4$ (ie, the root has $4$ daughter nodes and all other nodes have $...
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158 views

a variant of the Kleene tree

The (a?) Kleene tree is a computable (a.k.a. decidable) sub-tree of the full binary tree with no computable path. It is well-known. I need a variant. (For those in the know, I need a c-bar which is ...
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52 views

Is there a name for this variant of the MST and the TSP?

Suppose I am given a weighted graph $G$ that contains a "start vertex" $v_0$, and my goal is to construct a set of paths that all originate at $v_0$ and touch all of the vertices of $G$, with as ...
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1answer
176 views

A criterion for rooted trees to be isomorphic based on walks

Suppose you have two rooted trees $T_1$ and $T_2$ with roots $r_1$ and $r_2$, respectively. Furthermore, for every $k\ge 0$, the number of walks of $T_1$ starting at $r_1$ of length $k$ is equal to ...
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1answer
107 views

Does anyone have the correct link to treewidth.com?

Several posts (and this) on StackExchange sites like MO have some link-rot. For example, I've been looking into tree decomposition and keep coming across references to treewidth.com, but the link ...
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169 views

on counting the number of trees on Kn (case)

During my reasearch I have stumbled across a problem that can be presented in such way: "How many are there spanning trees on Kn such that every tree contains v: deg(v) = k, for a given k" The ...
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A separation property of graphs of bounded tree-width

The following separation property of trees is well-known and in fact easy to prove (see e.g. the paper "Covering a hypergraph of subgraphs" by Noga Alon, Lemma 2.2) Let $T$ be a tree and $r, m$ ...
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295 views

Embedding of real trees into $\ell_1(\Gamma)$

It seems plausible that any real tree or ${\mathbb{R}}$-tree in the sense of the definition in https://en.wikipedia.org/wiki/Real_tree admits an isometric embedding into the Banach space $\ell_1(\...
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LDP respectively almost sure convergence in the context of randomly weighted trees

I am currently working on the following Problem: Imagine you are given a $d$-ary tree $T_d$, which means an infinite tree with one vertex $x_0$ on top and in which each vertex has $d$ children. Next,...
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1answer
65 views

Billera Tree Space

I am studying the tree space of Billera and I do not really understand why it is an Hadamard Space. I have already read L. Billera, S. Holmes, K. Vogtmann, Geometry of the space of phylogenetic trees, ...
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1answer
173 views

How to uniquely define a tree? [closed]

In an undirected unlabled graph $G=(V,E)$, we want to find a tree as a subgraph, such that the graph can be decomposed into edge disjoint trees(all the tress are isomorphic). How to define such a tree ...
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1answer
286 views

Computing the probability of reaching any leaf of an $n$-ary infinite probability tree

Suppose you have two players $X$ and $Y$ fighting, both of which have $n\in \mathbb{N}, n\geq1$ life. Each player has a probability $p_i$ of doing $i$ damage for all $i\in[0, n]$. Note that $p_0$ is ...
4
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1answer
216 views

Counting trees according to endpoints

Question: Is there a nice (or any) formula for the generating function $$T(x,y) = \sum_{m,n} t_{m,n} x^my^n,$$ where $t_{m,n}$ is the number of trees with $m$ vertices and $n$ endpoints? ...
4
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1answer
349 views

The number of monotone full binary trees

Let $\rho$ be an equivalence relation on a semigroup $S$. A subsemigroup $S'$ of $S$ is called a $\rho$-cross-section of $S$, provided that $S'$ contains exactly one representative from each ...
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68 views

Is there a polynomial time algorithm for Poly-trees (Oriented trees) isomorphism?

In terms of graph isomorphism complexity classes Trees have a polynomial time algorithm and Directed Acyclic Graphs (DAG's) do not (so far). What about Poly-trees (oriented trees)? These are DAG's ...
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Is $\clubsuit_{\omega_1}$ enough to get Suslin tree?

This is problem 15.3 in Arnie Miller's problem list: (Juhasz) Suppose there exists $\langle A_{\alpha} : \alpha \in L \rangle$, where $L$ is the set of limit ordinals below $\omega_1$ and for each $\...
4
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1answer
191 views

Existence of $\kappa$-Suslin trees above a measurable cardinal

We have learned from Joel David Hamkins and Monroe Eskew that: Answers: Having a measurable cardinal $\delta$ we can force a $\kappa$-Suslin tree for many $\kappa$'s above $\delta$. But is the ...
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521 views

Which finite groups are not the automorphism group of some rooted finite tree?

The question is as given in the title: Which finite groups are not the automorphism group of some rooted finite tree? A rephrasing could be: Is any finite group representable as the automorphism ...
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246 views

Embedding Euclidean buildings into products of trees

A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.) Question: Is it true ...
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1answer
192 views

Determinant of the oriented adjacency matrix of a tree

Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries $$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\ -1 &...
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Solutions for n such that the path number to the n-th node is also n in a complete, rooted, ordered k-ary tree

The path to find the $n$-th node within a complete, rooted, ordered $k$-ary tree can be represented by a number to the base $(k+1)$ that contains no zero digits. See A245905 in OEIS as an example for ...
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546 views

Number of paths through infinite trees with given “growth rates”

(Preface: This may be a naive or easy question for experts....) Consider an infinite tree, rooted on the left, where each node has two children; the number of nodes at each level (distance from the ...
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1answer
305 views

Game on the tree [closed]

There's a problem from programming competition which already finished: http://codeforces.com/contest/458/problem/F Two weeks already passed but still nobody solved it yet - in fact you can see here ...
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3answers
556 views

Is there a formula for the number of labeled forests with $k$ components on $n$ vertices?

Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. My question is: Is there a generalization of this formula for forests? Let $f_{n,k}$ denote the number of ...
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2answers
313 views

High dimensional Steiner tree

Given n affinely independent points in n-1 dimensional Euclidean space, how is the minimum Steiner tree constructed? Or assuming that the topology of the Steiner tree is given, is there an easy way ...
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118 views

Growth in families of trees

I'm hoping that the question below is simple thermodynamic formalism, but I can't quite make it work. Any help would be very welcome. Let $\Sigma:=\{0,1\}^{\mathbb N}$ and let $\Sigma^*$ be the set ...
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Trees with a maximal convex hull: are the only optimal solutions Steiner trees?

For given $n\geqslant 3$, I'm looking for a connected set composed of $n$ equal segments in the plane such that the convex hull of it has maximal area $A(n)$. To simplify notation, we'll take $\dfrac{...
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More asymptotics for trees

This is a follow up to my recent question on the asymptotics of A003238. Lucia gave a fine answer to that question, but as I hinted the 'real' problem I have in mind is slightly different, and I've ...
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297 views

Is not SH + not CH consistent?

I guess that $\lnot$SH + $\lnot$CH is consistent, but I have not found this question discussed anywhere. Is there any relatively simple model of $\lnot$SH + $\lnot$CH?
3
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1answer
147 views

Ratio of expected diameter and height of a conditioned Galton-Watson tree

A Galton-Watson tree is the family tree of a Galton-Watson process. Let $T_n$ denote a Galton-Watson tree conditioned on total population size $n$. The time of extinction is its height $H(T_n)$ and ...
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1answer
168 views

Switching representatives of right coset in a paper on fundamental domain of tree of GL(2)

I have a question concerning the following paper: "The fundamental domain of the tree of GL(2) over the function field of an elliptic curve" by Shuzo Takahashi. (1993, Duke Math. J., Vol. 72, No. 1). ...
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276 views

Equality-preserving embeddings of finite trees

For finite trees $T_{1}$ and $T_{2}$ labelled by elements of some infinite set $S$, (we may assume that $S=\mathbb{N}$ without loss of generality), we define an equality-preserving embedding $f$ to be ...
2
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1answer
244 views

A variant of Kruskal's theorem

For $X$ and $Y$ finite sequences of finite trees, let us say that $X$ is everywhere contained in $Y$ ($X\subseteq_{ec}Y$) iff, for every $y\in Y$, there is some $x\in X$ such that $x$ is a minor of $y$...
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Can we have a $\kappa$-Suslin tree where $\kappa$ is above a measurable cardinal?

Question: Can we have a set theory in which there exists a $\kappa$-Suslin tree with $\kappa$ larger than the least measurable cardinal? A $\kappa$-Suslin tree is a tree with levels indexed by $\...
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109 views

Finding the number of leaf nodes at specific level of a random tree

Given a uniform recursive tree (URT) of size $N$ rooted at one node whereby the tree is generated as follows: Starting with a root node, at each iteration, a new node is connected to one of the ...