**6**

votes

**0**answers

76 views

### Upper bound for a sum including Andre polynomial coefficients

Let $ c_{n,k} $ be a sequence 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{0.5cm} (1 \leq k ...

**3**

votes

**1**answer

51 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, ...

**1**

vote

**1**answer

147 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 ...

**2**

votes

**1**answer

274 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**

votes

**1**answer

192 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?
...

**1**

vote

**0**answers

57 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 ...

**3**

votes

**1**answer

190 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 ...

**9**

votes

**1**answer

404 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 ...

**4**

votes

**1**answer

171 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

**1**answer

333 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 ...

**6**

votes

**1**answer

220 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 ...

**3**

votes

**1**answer

182 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 ...

**1**

vote

**0**answers

48 views

### 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 ...

**10**

votes

**2**answers

443 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 ...

**1**

vote

**3**answers

332 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 ...

**5**

votes

**0**answers

115 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 ...

**7**

votes

**2**answers

128 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 ...

**10**

votes

**1**answer

290 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 ...

**2**

votes

**2**answers

280 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**

votes

**1**answer

165 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). ...

**1**

vote

**1**answer

128 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 ...

**6**

votes

**1**answer

238 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 ...

**4**

votes

**2**answers

270 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**

votes

**1**answer

239 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 ...

**15**

votes

**0**answers

622 views

### Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$.
In this question I would like ...

**5**

votes

**2**answers

231 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 ...

**1**

vote

**0**answers

96 views

### Database of non-isomorphic trees

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

**2**

votes

**0**answers

82 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 ...

**7**

votes

**0**answers

126 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 ...

**7**

votes

**1**answer

402 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 ...

**5**

votes

**2**answers

287 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 ...

**13**

votes

**1**answer

516 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 ...

**8**

votes

**1**answer

152 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 ...

**2**

votes

**2**answers

162 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 ...

**9**

votes

**4**answers

588 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 ...

**3**

votes

**2**answers

491 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 ...

**9**

votes

**1**answer

336 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

**3**answers

348 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, ...

**3**

votes

**0**answers

78 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 ...

**7**

votes

**2**answers

751 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 ...

**1**

vote

**0**answers

216 views

### Is the automorphism group of a homogeneous (locally finite) tree unimodular?

I have seen somewhere (that I don't remember now) that the (full) automorphism group of a k-regular tree is unimodular. I assume a k-regular tree is the same thing as the homogeneous tree of degree k ...

**2**

votes

**0**answers

101 views

### Groups acting on non-locally-finite trees with independence and specified local actions

Suppose I have a biregular tree $T_{m, n}$ (not necessarily locally finite), with distinct cardinal numbers $m, n$, so Aut$(T_{m, n})$ acts on $T_{m, n}$ without inversion. Let $V_m$ be those vertices ...

**4**

votes

**0**answers

509 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 ...

**1**

vote

**2**answers

147 views

### Are trees spectrally determined?

Are trees (connected acyclic graphs) determinable up to isomorphism by their spectra or characteristic polynomials? If not, what other pieces of information may help determine the tree?

**2**

votes

**1**answer

262 views

### Hasse Diagrams of trees with height $>\omega^2$

I'm looking to improve my intuition and visualization of what large countable trees look like and I've ran into the issue that I have no understanding of what a tree of height, say, $\omega^2$ looks ...

**2**

votes

**1**answer

198 views

### Group actions on trees and translates under hyperbolic elements

I have the following question regarding group actions on trees to which I suspect the answer to be "yes", but it could very well be that extra conditions are required (it is certainly true for free ...

**3**

votes

**1**answer

166 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 ...

**8**

votes

**2**answers

228 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 ...

**2**

votes

**0**answers

814 views

### Matula-Goebel ordering of rooted trees intrinsic?

I was somewhat recently introduced to the Matula-Goebel bijection between rooted trees and natural numbers. (nicely illustrated here http://keithbriggs.info/matula.html) Looking through them, I ...

**11**

votes

**2**answers

488 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 ...