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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1answer
51 views

Mathematics of genealogical trees

I really searched a lot but did not find anything meeting my needs: A place where questions of genealogy, especially the structural and combinatorial analysis of genealogical "trees" of descendants ...
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0answers
33 views

What are the automorphisms of an $n$-regular tree?

Let $T$ be the connected tree in which each vertex has $n$ neighbors. (So $T$ is infinite.) What is the full automorphism group of $T$?
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0answers
40 views

Number of nodes with even offspring

I've been working on a combinatorics assignment, and while the last few questions had clever solutions which didn't involve functional equations and the use LIFT, I fear I'm at my end. Given a ...
2
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1answer
226 views

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 ...
2
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1answer
50 views

Number of upper sets of size $n$ in a finite tree

Consider a finite tree $T = (V, <)$, where $y < x$ means that $y$ is the parent of $x$. We assume that $T$ has a unique root $r$ that has no parent. An upper set of $T$ is a subset $S$ of $V$ ...
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0answers
74 views

Presentation of tree decompositions (and related concepts) in terms of continuous maps?

A tree decomposition of a graph $G$ is commonly defined in terms of a tree $T$ with the following structure: Each vertex $t \in V(T)$ is associated to a set $X_t \subseteq V(G)$; The union ...
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0answers
67 views

Groups acting on (regular) trees with finite quotient

Let $T$ be a regular tree, and suppose that $G \leq \mathrm{Aut}(T)$ has finite quotient graph, $T / G$. Is it true (in general) that $G$ will have trivial centralizer in the full automorphism group? ...
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618 views

Minimum Spanning Tree in a Complete Graph

We generate a complete euclidean graph by taking N random points from a limited (1.0 x 1.0 square) 2D space, connecting them all together (complete graph) and giving the edges weights proportional (or ...
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0answers
100 views

A few questions about a relationship between some integer sequences and infinite recursive trees

In his book Gödel, Escher, Bach Douglas Hofstadter defines the following two integer sequences: Hofstadter G-sequence: $a(n)=n-a(a(n-1))$ Hofstadter H-sequence: $a(n)=n-a(a(a(n-1)))$ He says ...
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0answers
90 views

How matroids can help me locating trees inside a graph?

Background I am working on a project at present involving graph analysis. I basically need to mathematically model trees inside my graph. How can this be done using Matroids? What I am looking for ...
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0answers
73 views

Embedding tree metric isometrically into $\ell_\infty$

I just started (independent) learning on metric embeddings from the Fall 2003 offering of the course at CMU. I have a limited mathematical background and alas, it made me stumble at the first exercise ...
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3answers
62 views

Why do the children of a node $n$ in a complete binary tree have indices $2n $ and $2n+1$?

The complete binary tree is breadth-first ordered 1 to $n$ where $n$ is the number of nodes. The thing I cant seem to understand is that why are the children of node $N$ always $2N$ and $2N+1$? For ...
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2answers
65 views

Can we construct from $[0,\omega_1)$ a space which is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

I have asked in here a question which tured out to make no sense. I think I have found the confusion and would like to try and rephrase my question: Let $E$ be a topological space, $q \in E$. ...
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5answers
8k 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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2answers
106 views

A bijection between ordered trees and ballot lists

Could you help me to prove that there is a bijection between ordered trees with $n+1$ leaves and ballot lists with $n$ A's and $n$ B's? A ballot list is a sequence of A's and B's such that all ...
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2answers
164 views

Is having no directed cycles in a directed acyclic graph (DAG) a byproduct of the design, or is it intentional?

According to Wikipedia: That is, it is formed by a collection of vertices and directed edges, each edge connecting one vertex to another, such that there is no way to start at some vertex v ...
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1answer
53 views

Largest order of automorphism group on a rooted tree?

MacArthur, Sanchez-Garcia, and Anderson have used the ratio of the order of $|Aut(G)|$ and $n!$ (i.e., order of $S_n$) as a normalized measure of the symmetries present in a graph. I am working on ...
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2answers
103 views

syntax tree for the word (())()()

I have to create the syntax tree for the word (())()() . That's what I have tried: Could you tell me if it is right?
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2answers
125 views

Trees with vertex set

I am having hard time understanding and solving the following question: There are exactly three trees with vertex set {1,2,3}. Note that all these trees are paths; the only difference is which ...
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1answer
161 views

Proofs involving subtrees of a tree

I have found some claims about trees in my graph theory text, and I am wondering if corresponding proofs can be found, as I cannot find any online or in another text. First, If $T_1$ and $T_2$ ...
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1answer
2k views

Number of Trees with n Nodes

I am struggling with a question that asks the number of trees that exist with x nodes and max level z. During my research I found that the number of binary trees with x nodes can be obtained by ...
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3answers
75 views

is MST a Steiner tree?

I am a little bit confused about MST and Steiner tree? Is an MST a steiner tree?? and suppose we are given a weighted undirected connected graph G = (V,E) and S ⊆ V is the smallest subtree of an MST ...
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2answers
161 views

Generating function for planted planar trees

I need your help to solve this problem : Give a generating function for planted planar trees with all degrees odd. Show that the number of such trees with $2k+1$ non-root vertices is ...
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1answer
36 views

Graph G with two Spanning Trees

Let's assume that Graph $G = <V,E>$ has two Spanning Trees $G_a = <V, T_1>$ and $G_b = <V,T_2>$ where $T_1 \cap T_2 = \emptyset$ and $T_1 \cup T_2 = E$. Prove that $\chi(G) \le 4$ ...
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2answers
313 views

single elimination tournament, don't understand question?

A single elimination tournament is performed in rounds. In each round the teams each play exactly one game and the winners continue, and the losers are knocked out of the competition. So, in each ...
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1answer
85 views

How to establish bijective between the set of rooted trees and natural numbers, using Godel numbering?

Consider the structure of a rooted tree independent of its underlying set, (i.e. in the sense of trees as combinatorial species). I know a number of ways which we can encode any such tree in natural ...
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3answers
836 views

Graph with cycles proof questions

Two questions I'm stuck with: If C is a cycle, and e is an edge connecting two nonadjacent nodes of C, then we call e a chord of C. Prove that if every node of a graph G has degree at least 3, then ...
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3answers
1k views

Sufficient conditions on degrees of vertices for existence of a tree

I am answering a question for an assignment, but I am not sure if my proof is valid, can someone look at it for me? the question: "there is a tree with $p$ vertices. If $d_1, d_2, \dots , d_p$ are ...
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2answers
63 views

Introductions to posets on algerbaic structures (Everything I need to know about them)

I need a good and complete introduction to Tree-like orders and partial orders on algebraic structures with one operations. I accept basic texts too. I'm looking for free online texts mostly because ...
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2answers
39 views

Proof that spanning trees can be converted into one another

Let $T_1$ and $T_2$ be two spanning trees of a graph $G$. Prove that if $e$ is an edge of $T_1$, there exists an edge $f$ in $T_2$ so $T_1-\{e\}+\{f\}$ also is a spanning tree. For $\{e\}\subset ...
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2answers
35 views

How to go from Tree to Total orders

Given a tree $T=(X,E)$, is it guaranteed for any orientation of the edges $E$, there exist a strict total order preserves it? For instance, let $X=\{x_1,x_2,..x_n\}$ and $E=(x_i,x_{i+1})$ the result ...
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2answers
78 views

How to prove that the smallest asymmetric tree has at least 7 vertices?

Find the smallest possible number of vertices an asymmetric tree can have (i.e. prove that no smaller tree can be asymmetric). I think that the answer is 7, but I don't know how to prove it.
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1answer
33 views

Number of binary trees with same X-order and Y-order

What is the number of binary trees which have the same X-order and Y-order as the given tree? Example: X-order: POSTORDER(T) = POSTORDER(TL) POSTORDER(TR) root Y-order: ANTIORDER(T) = ...
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2answers
1k views

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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4answers
829 views

How many labeled trees exist on n vertices with exactly 3 vertices of degree 1?

My combinatorics class is covering spanning trees right now and one of the questions being asked is "What is the number of labeled trees on n vertices with exactly $3$ vertices of degree $1$?" I've ...
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2answers
68 views

In a tree, is there always a sink where every longest path ends in?

Let $T$ be an undirected tree. Can we always find a leaf vertex $s$ such that every longest path of $T$ has its other endpoint in $s$? It's easy to see that every longest path passes through the ...
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2answers
93 views

Some equivalence relation from flipping binary trees

I know almost nothing in combinatorics, so this question might be very easy, or well-known. Fix a number $n$. We will consider rooted planar binary trees with $n$ leaves. We will distinguish between ...
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1answer
38 views

Connection Trees and Partition

In our lecture we just had a short excursion into the tree-world. But the professor mentioned some connection between Ramsey and König's Infinity Lemma (If $T$ is a tree of hight $\omega$ with all ...
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2answers
138 views

Draw Graph from distance to other nodes

I have a matrix that shows the distance from a node to another node: A B C D E A 0 2 4 3 1 B 2 0 2 1 3 C 4 2 0 2 1 D 3 1 2 0 2 E 1 3 1 2 0 To clearify: The 2 ...
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2answers
487 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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1answer
187 views

What is the length of the Minimum Spanning Tree

What is the length of the Minimum Spanning Tree for the following weighted graph? Solution. The length of any minimum spanning tree for this graph (and there is more than one) is 60. The graph and ...
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2answers
91 views

spanning trees of graphs

Assume we have a simple connected graph G, how would start a prove of the following statement? For any edge of G, there is a spanning tree of G that contans it. I have decided that this is a true ...
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1answer
137 views

For BSP generation, How to intersect or locate a triangle with a plane defined by another triangle?

I've hit a stumbling block in my project to draw the Utah teapot. I want to generate a binary space partition tree of a set of 3D triangles. The decision step in the recursive tree-construction ...
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1answer
36 views

If $G$ is a tree and $\forall v_1, v_2 : dist(v_1, v_2)$ is even, where $v_1, v_2-$ are leaves $\Rightarrow \exists!$ a maximal independent set.

If $G$ is a tree and $\forall v_1, v_2 : dist(v_1, v_2)$ is even, where $v_1, v_2-$ are leaves of the tree $\Rightarrow \exists!$ a maximal independent set. Give some clue please! Thanks anyway!
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1answer
165 views

Existence of a spanning tree with certain properties

Let $\Gamma$ be a finite, connected graph (multiple edges between two vertices are allowed). Fix a vertex $u_0\in V\Gamma$. Does there exist a maximal subtree (i.e., a spanning tree) $T\subset\Gamma$ ...
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1answer
206 views

Depth distribution of normalized decision trees?

Lets work with the following inductive definition of a decision tree: 1) $\bot$, $\top$ are decision trees. 2) If $x_i$ is a variable and $T_0$, $T_1$ are decision trees then $(\lnot x_i \land T_0) ...
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2answers
475 views

Graph - Minimum spanning tree

I have a graph with a cycle ($v_1,\ldots,v_k, v_1=v_k$). Claim: If there is a cycle with 2 edges of the same weight, and they are the heaviest edges in this cycle, then there is more than one Minimum ...
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1answer
23 views

Prove that every subgraph of forest has at least one vertex of degree < 2

So I know that a forest is a graph that has no cycles. This is what I had in mind: Assume that we have the subgraph T, which has two options: to be connected or not. if it's connected it has to be a ...
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2answers
52 views

When trees are the answer: what is the question?

For which optimization problems are (abstract) trees the best solution? E.g. binary search trees are somehow optimal data structures for quick search. But why for example do botanic trees grow as ...
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1answer
115 views

Number of rooted subtrees of given size in infinite d-regular tree

Currently I am reading a paper where the author states: [...] It is well-known that an infinite $D$-regular rooted tree contains precisely $\frac{1}{(D-1)u + 1} \binom{Du}{u}$ rooted subtrees of ...