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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Graph Algorithm and Cycle Detection

In $O(|V|+|E|)$, we can detect whether a Directed Graph has a cycle or not. ---> True In depth-first seach on DAG, there is no Back Edge. ---> True With known Number of Edges, in $O(|V|)$ and not ...
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43 views

Counting unlabeled and non-uniquely labeled trees

I recently learned about Cayley's formula, which states that the number of trees on $n$ labeled vertices is $n^{n-2}$. As I understand it, this works because we can prove that there are $n^{n-2}$ ...
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70 views

Examples of forcings which add no “definable” Aronjain tree

Maybe a bit board question but: Fixing a regular cardinal $\kappa$ in the ground model, I am looking for examples of set forcing notions which preserve regularity of $\kappa$ and add no new $\kappa$ ...
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42 views

Delete nodes that satisfy a property

I want to write a function that takes as argument a pointer A to the root of a binary tree that simulates a (not necessarily binary) ordered tree. We consider that each node of the tree saves apart ...
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37 views

How can I infer order from partially ordered discrete sequences?

A really interesting problem that I can't stop thinking about! Have run in to this a couple of times but yet to find a smart approach to either solve or frame this problem. This is my try at ...
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84 views

Can “tit for tat” strategy be defined in monadic second-order logic?

Prisoner's dilema game can be represented as a game tree, which could be infinite game with corresponding infinite game (binary) tree in common case. There is well-known tit for tat strategy, which ...
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1answer
66 views

Shortest path between two vertex

How we can find Shortest path between two vertex in a weighted directed acyclic graph that has positive and negative weight. in O(|V|+|E|)? thanks to all.
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1answer
80 views

Width and height of binary tree is $\theta(n)$?

we know this definition: Given a binary tree, Width of a tree is maximum of widths of all levels. Let us consider the below example tree. ...
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1answer
66 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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38 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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1answer
55 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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90 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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71 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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763 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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107 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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94 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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79 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
74 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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1answer
341 views

How to convert parentheses notation for trees into an actual tree drawing?

Trees are usually drawn as a set of objects connected by edges. But sometimes one sees a non-graphical, parentheses-based notation, like on the example below. What does the indentation mean in such ...
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1answer
79 views

Number of spanning trees for these 2 figures

The solution to the number of spanning trees of the graph below is given by $6$ and $4 \times 4 - 1$ for Graph A and B respectively. I'm not sure how to get this. Please assist. I did ask a similar ...
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1answer
132 views

Number of spanning trees of this graph

The solution to the number of spanning trees of the graph below is given by $3 \times 2 \times 3 = 18$. I'm not sure how to get this. Please assist. Thanks! Notes: Just in case anyone was ...
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2answers
72 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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1answer
3k 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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2answers
108 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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175 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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26 views

Trees-related proof

I just began my Graph Theory course, so I'm pretty knew in this area, at least when it's about formal proofs(I have some experience on intuitive level, implementing certain algorithms related to graph ...
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3answers
42 views

Traversing through a binary tree

Consider a full binary tree of n nodes numbered from 1 to n in the common top-down left-to-right manner. For the sake of the ...
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1answer
89 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
196 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
200 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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3answers
1k 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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49 views

Rooted Tree and Greedy Algorithms

In a Rooted Tree, we have a message on Root. in each step, each node that has a one copy of message, can transfer this message to at most one of it's childeren. we want to use minimum step and send ...
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40 views

The number of (non-equal) forests on the vertex set V = {1, 2, …,n} that contains exactly 2 connected components is given by

The number of (non-equal) forests on the vertex set V = {1, 2, ...,n} that contains exactly 2 connected components is given by $\sum_{k=1}^{n-1} {n-1 \choose k-1} k^{k-2} (n-k)^{n-k-2}$. I am unsure ...
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1answer
30 views

Existence of infinite subsequence of trees with a subtree contained in the sequence

Assume a statement: For every infinite sequence of rooted trees $\{T\}_{i=0}^\infty$ there is an index $j\geq0$ such that there are infinitely many trees in $\{T\}_{i=0}^\infty$ which contains ...
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3answers
97 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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195 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
42 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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541 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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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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1answer
126 views

Prove that a graph cannot have two distinct spanning trees

I'm confused with this proof. More so that I think I'm confused as what distinct in this context means? Initially I thought it was that these 2 possible spanning trees cannot share the same edges, but ...
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1answer
15 views

Proof of a tree with a vertex of degree k and less than k vertices of degree 1

The question is : Does there exist a tree with a vertex of degree k and less than k vertices of degree 1? I tried a lot but it is impossible to find. There is no tree with a vertex degree k and less ...
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1answer
60 views

A decision tree has an expected depth of at least $\log n!$

I am looking at the proof of the following theorem and I have some questions. The theorem is the following: On the assumption that all permutations of a sequence of $n$ elements are equally ...
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3answers
90 views

Proof of an $\iff$ statement on binary trees

Let $x$ and $y$ be two nodes of a binary tree $B$. Prove that $x$ is an ancestor of $y$ $\iff$ $x$ stands before $y$ in the pre-order traversal of $B$ and $x$ stands after $y$ in the ...
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2answers
47 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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59 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
170 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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40 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
89 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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1answer
40 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
193 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 ...