For questions about trees in graph theory, which are connected graphs with no cycles. Also can be used for questions about forests, which are graphs that are disjoint unions of trees.

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$\kappa$-Suslin (Aronszajn, Kurepa) subtree of the complete binary tree, $2^{\lt \kappa}$

This is from chapter 2 of Kunen, Set theory: an introduction to independence proofs. Given $\kappa$ a regular cardinal, and the existence of a $\kappa$-Suslin (Aronszajn, Kurepa) tree, show that ...
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2answers
103 views

Computing Ancestors of # for Stern-Brocot Tree

Reading about the Stern-Brocot tree, the article gives this example: using 7/5 as an example, its closest smaller ancestor is 4/3, so its left child is (4 + 7)/(3 + 5) = 11/8, and its closest ...
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1answer
178 views

Exercise 1.1 in Serre's trees

I have in fact become stuck by the very first problem in Serre's book on Trees. It is a little bit embarrassing but ho-hum. I start with Serre's definition of direct limits. Let $(G_i)_{i \in I}$ be ...
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2answers
1k views

Show that a graph has a unique MST if all edges have distinct weights [duplicate]

Prove that if all edge-costs are different, then there is only one cheapest tree (minimum spanning tree or MST). (Use contradiction and make sure to keep track of the costs of the different trees ...
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68 views

Why is the intersection of countably many homogeneously Suslin subsets of $\,^{\omega} \omega$ homogeneously Suslin?

I found this assertion in these notes: The derived model theorem (Steel) right in the beginning on page 3, together with the remark that this is 'not too hard to show'. Unfortunately, I'm ...
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25 views

Simple criteria to know if the p-nary notation of an integer can generate a tree by preorder traversing?

I am treating with a preorder tree traversal structure(which means sequences where the children of each tree node are listed behind it) now for some other problems and the structure is like: ...
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1answer
53 views

Expected number of subtree removal in a tree.

I was solving this problem. In a gist the problem is as follows: You are given a rooted tree. On each step you choose a node randomly and remove the subtree rooted by that node and the node ...
4
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1answer
76 views

Infinite sequence of trees that are not subgraphs to each other

This is from a set of exercises and I am stuck to this. Please, have in mind, that I want to understand how it's solved, I am not just looking for a solution. Define an infinite sequence of trees ...
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2answers
5k views

Finding number of homeomorphically irreducible trees of degree N

There is a scene in Goodwill Hunting where professor challenges students with task of finding all homeomorphically irreducible trees of degree 10. This is discussed in many places, such as here and is ...
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346 views

Algorithm for generating homeomorphically irreducible trees of size n

In this video they talk about generating all the homeomorphically irreducible trees of size 10. I was wondering if there is a generating algorithm for generating all the homeomorphically irreducible ...
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219 views

Number of spanning arborescences

I am trying to prove the following result from my book: Let $G$ be a directed graph with vertices $x_1,x_2,\dotsc, x_n$ for which a directed Eulerian circuit exists. A spanning arborescence with ...
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104 views

Identify this combinatorial construction

I am no combinateur, but I stumbled across the following construction when studying an operad arising from information theory (actually it's a special algebra of an A$_\infty$-operad). It looked ...
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133 views

Are almost all rooted trees asymmetric?

It's well known that almost all graphs are asymmetric (have trivial automorphism group) and that almost all free trees are symmetric. By which argument do I see whether almost all rooted trees are ...
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5answers
281 views

Halting probability of random tree-generating algorithm

Suppose I have a tree-generating algorithm as follows. Begin with one root vertex. With equal probability, create either three subvertices or none. Recurse and repeat for each of the subvertices (if ...
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5answers
37k views

The maximum number of nodes in a binary tree of depth $k$ is $2^{k}-1$, $k \geq1$.

I am confused with this statement The maximum number of nodes in a binary tree of depth $k$ is $2^k-1$, $k \geq1$. How come this is true. Lets say I have the following tree ...
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5answers
8k views

How many edges does an undirected tree with $n$ nodes have?

How many edges does an undirected tree with $n$ nodes have?
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2answers
251 views

König's Infinity Lemma and Aronszajn Trees

I am working through the notes of my Set Theory lecture. There my professor wrote: 'Is there an uncountable $\kappa$ such that König's Infinity Lemma holds for $\kappa$? There are models where ...
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2answers
398 views

In any tree, what is the maximum distance between a vertex of high degree and a vertex of low degree?

In any undirected tree $T$, what is the maximum distance from any vertex $v$ with $\text{deg}(v) \geq 3$ to the closest (in a shortest path sense) vertex $y$ with $\text{deg}(y) \leq 2$? That is, $y$ ...
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256 views

Find a Generating Function for Ordered Rooted Ternary Trees

The Full Question If we let $T=$ the family of rooted ternary trees, $t_n =$ be number of trees in $T$ with $n$ nodes and $T(x) = \sum\limits_{n=0}^{\infty}w_nx^n$ be the generating function of $T$. ...
3
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1answer
60 views

Number of spanning trees of a labeled graph

This labeled graph is given, I need to find the number of its spanning trees. The number of spanning trees of the following graph is 3 and the number of spanning trees of this is 8 So as a ...
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2answers
83 views

Number of undirected trees

Given n numbered vertices I want to know the number of different trees that can be created with them. I know that cayley's theorem says it's $n^{n-2}$, but why can't it also be: ...
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3answers
357 views

Cheapest spanning tree

I am trying to prove the following: Let $x_1$ be any vertex of a weighted connected graph $G$ with $n$ vertices and let $T_1$ be the subgraph with the one vertex $v_1$ and no edges. After a tree ...
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3answers
199 views

How to calculate the expected maximum tree size in a pseudoforest

I would like to calculate the expected maximum tree size in a randomly generated pseudoforest of $N$ labelled nodes where self-loops are not permitted. Empty and single-node trees are also not ...
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2answers
588 views

Show that if G is a simple graph with at least 4 vertices and 2n-3 edges, it must have two cycles of the same length.

For $n\ge4$, let G be a simple n-vertex graph with at least $2n - 3$ edges. Prove >that G has two cycles of equal length. (West's Introduction to Graph Theory Q 2.1.42) I am trying to prove the ...
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2answers
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Can Prims and Kruskals algorithm yield different min spanning tree?

In this problem I am trying to find the min weight using the Prims and Kruskals and list the edges in the order they are chosen. For Prims I am getting order ...
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1answer
166 views

Virtually infinite cyclic groups act on a tree

A virtually infinite cyclic group $G$ is quasi-isometric to $\mathbb{Z}$ and thus has two ends; by Stallings theorem, $G$ acts (without inversion) on a tree with finite edge-stabilizers. But the ...
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2answers
266 views

Why for number of leaves in a tree (all types of trees) is it true

I have to prove the following claim, given the tree $T=(V,E)$, $|V|\geq3$: $$|V_1| \leq \frac { |V| \times (\Delta (V) - 2) + 2 }{ \Delta (V) - 1 } $$ where $|V_1| - $ number of leaves in a tree, and ...
3
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97 views

Proofs involving some general formulae for trees and binary trees.

So here I have 3 tree-related questions. 1) Let $n\geq2$ and let $d_1 ≤d_2 ≤···≤d_n$ be a sequence of integers. Show that there is a tree with degree sequence $d_1,d_2,...,d_n \Leftrightarrow \sum ...
3
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3answers
810 views

In binary tree, number of nodes with two children when number of leaves is given

For a binary tree what is the number of nodes with two children when the number of leaves is 20? I know that for complete binary tree, when the number of leaves is x then the number of internal nodes ...
3
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1answer
42 views

Height of quasi-complete binary tree

Let us define a quasi-complete binary tree as a rooted binary whose nodes have all two children except at most those of the penultimate level, which can have either one or two children. I read that ...
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1answer
68 views

Is this random binary tree finite?

Consider the following procedure for generating a random binary tree: Starting with a full binary tree (i.e., each node has either two or no children) we iterate over the leaves and (independently) ...
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114 views

How to call a tree with a single branch?

How do you call a tree with only one branch (in other words, where every vertex has maximum one direct successor)?
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1answer
5k views

How to draw all nonisomorphic trees with n vertices?

I have a textbook solution with little to no explanation (this is with n = 5): Could anyone explain how to "think" when solving this kind of a problem? (for example, drawing all non isomorphic ...
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2answers
2k views

Gallery of unlabelled trees with n vertices

Can anyone point me to a gallery (printed or online) of unlabelled trees, sorted according to their order (i.e., number of vertices)? That is, for each order n in oeis.org/A000055 (up to maybe n=11 ...
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2answers
436 views

Bijection between binary trees and plane trees?

I would like to describe a bijection between binary trees and plane trees. A binary tree has a root node and each node of the tree has at most 2 children (left and right). A plane tree has a root node ...
3
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1answer
562 views

Spanning Trees of the Complete Graph Avoiding a Given Tree

EDIT: I think everyone understood, but I never explicitly stated that I am looking at labeled spanning trees. Let $T$ be a tree contained in $K_n$ (the complete graph on $n$ vertices). How can one ...
3
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1answer
43 views

How many ways can I connect labeled trees into a tree.

Suppose I have the labeled trees $T_{1}, \ldots, T_{n}$ with $b_{1}, \ldots, b_{n}$ vertices respectively. I would like to know how many ways I can compose a tree from these trees by using all trees? ...
3
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1answer
208 views

Generating function for vertices distance from the root in a planar tree

I need you help to solve this problem: Consider a planar tree with $n$ non-root vertices. Give a generating function for vertices distance $d$ from the root. Proof that the total ...
3
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1answer
334 views

A Graph as a Union of K forests.

I want to show that a graph G that is a union of k forests has a chromatic number of at most 2k. I have narrowed my problem down to trying to show that any graph G that is a union of n trees has a ...
3
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1answer
263 views

Cubic (3-regular) graph spanning tree

Considering loop free cubic graphs (graphs where every node has 3 neighboring nodes): Is is possible to construct a spanning tree that only has nodes with 3 neighbors in the spanning tree or 1 ...
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2answers
127 views

Bounds on how far away most leaves are from the average height of a binary tree

I'm wondering if anyone can help me prove (or disprove) this statement? Say there is a rooted full binary tree (each non-leaf node has exactly two children) with a height of $h$, an average height ...
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1answer
49 views

How many trees on N vertices have exactly k leaves?

I need help on the topic of counting labeled trees (with its nodes numbered from 1 to N) with exactly k leaves. I have thought about surjective functions that return the father of a node, but I'm not ...
3
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1answer
53 views

Automorphism of Tree

Let $\sigma$ and $\theta$ be two automorphisms of tree $X$. I want to show that min$_{v\in V(X)}d(v,\sigma(v))=$min$_{v\in V(X)}d(\theta^{-1}\sigma\theta(v),v)$. I know every automorphism of tree is ...
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2answers
122 views

How many vertices does this tree have?

Suppose that $T$ is a tree. It has $e$ edges and $n$ vertices, and $\overline{T}$ has $10e$ edges. What is n? I think $n = 1$ is a solution, because $T$ can have no edges then, so $0=10*0$. A ...
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411 views

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non-trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as ...
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Number of binary trees with N nodes

I am trying to calculate the number of trees (non isomorphic) with n nodes (total including leaves). I think that there are n! such trees, but I don't know how to prove that. I know that the number ...
3
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1answer
162 views

Prong Corollary, $G$ has a subgraph isomorphic to $T$

There is a corollary in Diestel textbook Graph Theory. Corollary 1.5.4. if $T$ is a tree and $G$ is any graph with $\delta(G) \geq |T|-1$, then $T \subseteq G$, i.e. $G$ has a subgraph isomorphic ...
3
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1answer
94 views

What's the rank of this well founded relation?

Definition A tree is an ordered list of trees. (N.B these are finite objects and there is a very simple computable bijection of them with $\mathbb N$) Examples [] and [[],[],[]] and ...
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156 views

Tree having no vertex of degree 2 has more leaves than internal nodes

If $T$ is a tree having no vertex of degree 2, then $T$ has more leaves than internal nodes. Prove this claim by a) induction, b) by considering the average degree and using the handshaking lemma. ...
3
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1answer
88 views

Number of spanning trees in a complete split graph

A graph is a complete split graph if we can partition it into an independent vertex set and a clique, such that every vertex of the independent vertex set is adjacent to every vertex in the clique. ...