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.

learn more… | top users | synonyms

2
votes
1answer
171 views

Probability of passing through 3 specific nodes along a binomial tree

Consider a re-combining binomial tree with probability of up = $p$ and probability of down = $(1-p)$. Let $n$ be the number of time steps in the binomial tree (the $x$-axis is time, and each column of ...
2
votes
1answer
149 views

Finite Rooted Binary Trees

I am new to learning about finite rooted binary trees. This lemma below is from John Meiers book: Groups, Graphs and Trees. There is no aval proof in the book. I was just wondering is I could catch a ...
2
votes
1answer
635 views

Oriented trees and ordered trees

I have this confusion regarding ordered and oriented trees. I know they are both rooted and in ordered trees, the order is important. So lets say I have four nodes 1,2,3,4 then it is given that the ...
2
votes
1answer
95 views

Showing two recurrences to be identical

I am trying to prove Cayley's formula for number of labelled trees on n vertices using multinomial coefficients. The multinomial coefficient satisfies the recurrence: $\tbinom{n}{r_1,\cdots ...
2
votes
2answers
449 views

Number of undirected trees with labeled edges, one repeating

I need to find the number of undirected trees on $n$ vertices such that the edges (and not the vertices) are labeled and exactly one label appears twice (i.e. there are $n-2$ possible labels and they ...
2
votes
1answer
35 views

What is the “true” minimum spanning forest of a connected graph?

Normally, a minimum spanning forest of a graph G is defined as the union of minimum spanning trees of each of its components. This definition is a generalization of the minimum spanning tree of a ...
2
votes
0answers
72 views

Tree decomposition by hand for understanding

I am implementing "algorithm 2" from the paper "Treewidth computations I. Upper bounds" by Bodlander and Koster[1,page5] and I am not sure if I understand it or not. As I understand, the algoritm ...
2
votes
1answer
24 views

Finding a node in a full binary tree: expected number of comparisons

Consider a full binary search tree of height $k$ (the root is on level $1$ and the leaves on level $k$). By full I mean that all leaves are on level $k$ and level $k$ has exactly $2^{k-1}$ leaves. In ...
2
votes
0answers
63 views

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 ...
2
votes
0answers
47 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 $n^{n-2}$ ...
2
votes
0answers
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 $\kappa$ ...
2
votes
0answers
43 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 ...
2
votes
0answers
39 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 ...
2
votes
0answers
88 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 ...
2
votes
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.
2
votes
1answer
96 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. ...
2
votes
1answer
70 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 ...
2
votes
0answers
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 $T$?
2
votes
1answer
56 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 $V$ ...
2
votes
0answers
92 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 ...
2
votes
0answers
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? ...
2
votes
0answers
802 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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 ...
2
votes
0answers
82 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 ...
1
vote
3answers
75 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 ...
1
vote
1answer
364 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 ...
1
vote
1answer
80 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 ...
1
vote
1answer
136 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 ...
1
vote
1answer
159 views

Proof for binary tree is a planar graph

Suppose G is a binary tree. Is G necessarily planar? Give a proof, or a counterexample. My guess is that it is indeed planar but I am struggling to find a formal proof for this. EDIT: Is there a ...
1
vote
2answers
73 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, $q \in E$. ...
1
vote
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 ...
1
vote
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 ...
1
vote
2answers
178 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 ...
1
vote
2answers
42 views

solve recurrence relation: comparisons to construct binary search tree with maple

I would like to solve the recurrence relation for the average number of comparisons necessary to the construction of a binary search tree. the recurrence is $$ i(n) = n - 1 + \frac{2}{n} ...
1
vote
2answers
28 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 ...
1
vote
3answers
45 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 ...
1
vote
1answer
93 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 ...
1
vote
2answers
215 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 ...
1
vote
1answer
213 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 $T_2$ ...
1
vote
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 ...
1
vote
2answers
54 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 ...
1
vote
3answers
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 ...
1
vote
1answer
36 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 ...
1
vote
3answers
100 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 ...
1
vote
2answers
199 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 ...
1
vote
1answer
43 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 $\chi(G) \le 4$ ...
1
vote
2answers
555 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 ...
1
vote
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 ...
1
vote
1answer
54 views

Existence of infinite subsequence of trees with a special condition

For rooted trees, define $children(v)$ as the number of children of the vertex $v$. Assume two operations on rooted trees: contract an edge: choose an edge $E$, join two vertices adjacent to $E$ ...