**7**

votes

**2**answers

632 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 ...

**4**

votes

**1**answer

242 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 ...

**15**

votes

**0**answers

591 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 ...

**7**

votes

**0**answers

120 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 ...

**5**

votes

**0**answers

110 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

**0**answers

461 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

**0**answers

64 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 ...

**2**

votes

**0**answers

54 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 ...

**2**

votes

**0**answers

93 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 ...

**2**

votes

**0**answers

608 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 ...

**1**

vote

**0**answers

46 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 ...

**1**

vote

**0**answers

94 views

### Database of non-isomorphic trees

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

**1**

vote

**0**answers

179 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 ...

**0**

votes

**0**answers

22 views

### Merging regions of function with similar mean and deviation using statistical test

I have got a question related to statistical tests that I would like to use in a new algorithm I am developing. Given an action space $x$, the algorithm would identify the regions in the function ...