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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27
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9answers
3k views

Is the empty graph a tree?

This is a boring, technical question that I stumbled upon while making a contribution to Sage. I would still like to hear a constructive answer so hopefully the question does not get closed. The ...
22
votes
2answers
1k views

Does this poset have a unique minimal element?

Recently I have been thinking about the following poset: the underlying set is $\mathcal{AFT}$ consisting of all (finite) automorphism-free undirected trees (with at least one edge to exclude the ...
13
votes
5answers
908 views

Is the following invariant of rooted trees a complete invariant?

Recall that rooted trees may be generated by starting with a trivial rooted tree (just a vertex), along with the operations of grafting a number of trees (identify their roots) and adding a new vertex ...
13
votes
1answer
536 views

Are the asymptotics of A003238 known?

Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins 1, 1, 2, 3, 5, 6, 10, 11, 16, ... and it is ...
11
votes
2answers
514 views

Series defined by a fixed-point functional equation

Description I my work, I met fixed-point functional equations of a very specific form. Since I am not expert in the domain of functional equations, I have not pushed the study of these very far. Here ...
10
votes
2answers
540 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 ...
10
votes
1answer
301 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 ...
10
votes
2answers
867 views

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 ...
9
votes
4answers
631 views

Are there two computable binary trees such that each has a branch not computing any branch through the other?

It is a well-known elementary classical result in computability theory that there are computable infinite binary trees $T\subset 2^{<\omega}$ having no computable infinite branch. (One can build ...
9
votes
1answer
516 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 ...
9
votes
1answer
418 views

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 $\...
9
votes
1answer
347 views

Suslin trees in ccc ideals

A countably complete ideal $I$ on a set $Z$ ideal is c.c.c. when there is no uncountable family of pairwise disjoint $I$-positive subsets of $Z$. If such an ideal exists, then there exists a weakly ...
9
votes
3answers
363 views

Characteristic polynomials of trees and E8

In thinking about constructing manifolds via surgery or plumbing, the following combinatorial problem comes up: If T is a tree with adjacency matrix A and I is the identity matrix of the same order, ...
9
votes
1answer
350 views

What is the precise relationship between “prodsimplicial sets” and rooted trees?

In Keven Walker's answer to the question, Cubical vs. simplicial singular homology it is written: Personally, I think it is more convenient to do singular homology with the larger collection ...
9
votes
1answer
163 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 &...
9
votes
0answers
256 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....
8
votes
1answer
154 views

Does being special on a club imply being special?

Let $T$ be an Aronszajn-tree, $C\subset \omega_1$ a club set and $f:\bigcup\limits_{\alpha\in C}T_\alpha\longrightarrow \mathbb Q$ a strictly increasing function (where $T_\alpha$ is the $\alpha$-...
8
votes
2answers
235 views

Distortion of tree embedding in Alexandrov spaces

It is a well-known theorem first proved by Bourgain that any map $\varphi:T_n\to H$ from the binary tree of height $n$ to a Hilbert space has distortion at least $C \sqrt{\ln n}$ where $C$ is a ...
7
votes
2answers
134 views

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{...
7
votes
1answer
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 ...
7
votes
1answer
410 views

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 ...
7
votes
0answers
77 views

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 $r, m$ ...
6
votes
1answer
293 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(\...
6
votes
1answer
244 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 ...
6
votes
1answer
137 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. ...
6
votes
2answers
157 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 ...
5
votes
2answers
253 views

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 $\...
5
votes
2answers
313 views

Average number of distinguished leaves in a binary tree

By a binary tree, I mean in this question a full rooted binary tree in which left and right child are labeled. A leaf of such a tree is a vertex of degree at most 1 (most references would probably ...
5
votes
2answers
273 views

Number of trees with the same matching number

Let $\sigma(n,m)$ be the number of trees with $n$ vertices $\{ v_1, \dots, v_n \}$ such that the matching number (the size of a maximum matching) is $m$. I have been trying to compute the value of $\...
5
votes
0answers
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 ...
4
votes
1answer
211 views

A distinguishing node property in trees?

Consider a tree with k nodes and for each node v the vector lv = (lv0, lv1, ..., lvk-1) with lvd the number of leaves (!) with distance d to v. I wonder whether two nodes v, w with lv = lw are ...
4
votes
2answers
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 ...
4
votes
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
votes
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 ...
4
votes
1answer
347 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 ...
4
votes
0answers
100 views

Using Crump-Mode-Jagers processes to get logarithmic bound on a random tree height

I am currently pursuing my PhD degree and in my research I came across a family of random trees. I need to prove a logarithmic asymptotic bound for the heights of such trees as their size grows. I ...
4
votes
0answers
568 views

Questions about dessin d'enfants, trees and their Shabat polynomials

This will be a series of questions, a few of which have been troubling me for quite a while now. Before I jump right in, let me first introduce a few notions which I will assume. (Note: All of these ...
3
votes
2answers
548 views

A structure of the group of automorphisms of an infinite binary tree

My friend asked me to ask his question here. Where he can find (a paper or a book) containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary ...
3
votes
1answer
175 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 ...
3
votes
2answers
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 ...
3
votes
1answer
170 views

Correspondence between fractal sets and trees

In Hillel Furstenberg's series lectures on ergodic theory in fractal geometry, he mentioned his search on finding a one-to-one correspondence between fractal sets and trees, however, I couldn't not ...
3
votes
2answers
276 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 $t(L)$...
3
votes
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, ...
3
votes
1answer
145 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 ...
3
votes
2answers
689 views

Fixed points of a group-operation on a tree, Serre's book “Trees” 6.3.4. and Prop 27

Hello! I have a problem with the following Lemma, which is mentioned in Serre's book "Trees" on page 60. In the book it is the Example 6.3.4.: Lemma: Let $G$ be a group acting (without inversion) on ...
3
votes
1answer
191 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 &...
3
votes
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). ...
2
votes
2answers
293 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?
2
votes
2answers
170 views

Normal subgroups of automorphism group of infinite cubic tree

Let $T$ denote the unique (countably infinite) tree in which every vertex has degree $3$ and let $G=Aut(T)$ be the group of graph automorphisms of $T$. I'm interested in the properties of this group,...
2
votes
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 ...