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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81 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 ...
0
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2answers
505 views

Proving a simple connected graph is a tree if adding an edge between two existing vertices of T creates exactly one cycle

When proving a simple connected graph is a tree if adding an edge between two existing vertices of T creates exactly one cycle, is it sufficient to just remove that edge that created a cycle, then it ...
3
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0answers
272 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 ...
3
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1answer
57 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 ...
2
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3answers
360 views

Graphs: trees, induction proof

I was wondering if you could help me prove the following. $G$ is a tree $\iff$ deleting any edge will disconnect it. And a similar one: $G$ is a tree $\iff$ adding any edge will create a cycle. I ...
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2answers
351 views

Confusion related to a graph problem

I have this question related to this graph problem Suppose that an n-node undirected graph G = (V , E) contains two nodes s and t such that the distance between s and t is strictly greater than n/2. ...
3
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1answer
111 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
597 views

“Ballot numbers” sum up to Catalan numbers

Summing certain numbers and comparing the results with OEIS, I found that $ \sum_{k=1}^n \frac{k^2}{n} \binom{2n-k-1}{n-1} = C_{n+1} - C_{n}, $ where $C_n$ denotes the $n^{\textrm{th}}$ Catalan ...
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2answers
137 views

Question about trees and generalizing the Principle of Dependent Choices.

One form of the Principle of Dependent Choices is that for any tree $T$ of height $\omega$ such that every node of $T$ has a successor, there is a branch of $T$ of length $\omega$. In this post, I ...
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0answers
89 views

Straight skeleton is a tree

Can anybody give me a hint on how to prove that the straight skeleton of every polygon is a tree. Here is the definition of the straight skeleton (taken from Wikipedia): The straight skeleton of a ...
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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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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0answers
43 views

Enumeration of symbols in grammatical expressions or vertices in tree graphs

I have expressions (type of a function) like e.g. $$f:(A\to B)\to C \to (D\to E)\to F.$$ (Where I understand $A\to B\to C$ as $A\to (B\to C)$, in case that is relevant.) There might be information ...
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2answers
316 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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0answers
272 views

Certain permutations of the set of all Pythagorean triples

The fact that the set of all primitive Pythagorean triples naturally has the structure of a ternary rooted tree may have first been published in 1970: http://www.jstor.org/stable/3613860 I learned ...
0
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1answer
121 views

Computational Complexity of Parallel Algorithms

Does parallelism factor in when deriving the computational complexity of a parallel algorithm? Suppose I have a perfect binary tree $T$ with leaves numbered $1$ to $n$, and an algorithm ...
2
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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 ...
2
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1answer
70 views

A way to codify (pre-calculatate) if a one Tree Node is a descendant of another

I have a simple, 1-directional tree representing the veins in a human body. It looks somewhat like this (red dots are nodes, blood flow is always downwards, sorry for my drawing): What I need is a ...
0
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1answer
308 views

Is this formula for the number of nodes for a complete tree or a full and complete tree?

In a lecture it was said that "How many nodes are there in a complete k-ary tree with height h?" and this was the answer: $$ \sum^{h}_{i = 0}k^i $$ where h is the height and k is the max number of ...
3
votes
2answers
1k views

What does lg x mean? is it $\log_2 x$ or $\log_{10} x$ in binary trees

I'm a bit confused, $\log_{10} x = \log x $ right? I believe I've read somewhere that $\log_{2} x = lg x$ but some people say lg = $\log$. So what does lg really stand for? specifically when talking ...
3
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2answers
89 views

Real tree and hyperbolicity

I seek a proof of the following result due to Tits: Theorem: A path-connected $0$-hyperbolic metric space is a real tree. Do you know any proof or reference?
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2answers
1k views

Proof by induction and height of a binary tree

I need some help with a simple proof. I want to know if this proof is correct: Let's define the height of a binary tree node as: 0, if the node is a leaf 1 + the maximum height of the children ...
3
votes
1answer
61 views

Tree formalisms

The intuitive notion of a tree in mathematics is quite straightforward. However, there are several different formalisms of the tree concept. The link http://ncatlab.org/nlab/show/tree lists several ...
4
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1answer
207 views

characteristic polynomial of the adjacency matrix of a tree

I have read that if $A$ is the adjacency matrix of a tree $T$, then we have that $$\det(\lambda I - A) = \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k N_k(T) \lambda^{n-2k} $$ where $N_k(T)$ is the number ...
0
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1answer
203 views

Relationship between ordered and binary trees

I am looking for a formula for the number of ordered trees with $n$ vertices and $l$ leaves as well as for a formula for the number of binary trees with $l$ left and $r$ right children. Finally, I ...
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2answers
272 views

Breadth first search tree's cycles [duplicate]

Possible Duplicate: Proof related to breadth first search I'm trying to prove the following: Suppose a connected graph $G$ has a cycle $C$ of length $n$. Prove that in any breadth-first ...
5
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2answers
244 views

Suppose there are two different spanning trees for a simple graph. Must they have an edge in common?

My instinct is yes, but I don't know how to formalize it into a proof. I still haven't wrapped my head around spanning trees yet. Any thoughts are appreciated!
0
votes
1answer
373 views

How to find the maximum number of vertices in a tree with respect to maximum path length and maximum degree value

Given a tree, find the maximum number of vertices $v$ in that tree using the maximum path length $p$ and a maximum degree that applies to all vertices $d$. Assuming that I drew my test tree ...
3
votes
2answers
228 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 ...
4
votes
1answer
4k views

Determining if two graphs are isomorphic

The answers are already given after the $=$ sign of each question. But I don't know how they arrived at these answers. What does it mean to say $f(A)=1$ and so on? I can't find the connection.
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1answer
148 views

Depth first search on graph

I have a homework problem I think I know the answer to, but want to double check Consider the graph with three nodes, $a$, $b$, and $c$, and the two arcs $a \rightarrow b$ and $b \rightarrow c$. ...
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0answers
47 views

What is the runing time of this algorithm involving length and depth?

I'm hoping that someone can shed some light on this running time. I have a "tree", for lack of a better description, that has a length $l$ and depth $d$. I want to maximize the tree size, which ...
2
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1answer
264 views

Length of Shortest Path in a Generated Binary Tree

Let's say I have a binary tree of $2$-tuples of positive integers starting with $(1,1)$. The left-child of any element $(A,B)$ is $(A,A+B)$, and the right-child of any element is $(A+B,A)$. Hence, the ...
1
vote
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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1answer
587 views

M-ary tree problem

A full $m$-ary tree $T$ has 81 leaves and height 4 1) Give the upper and lower bounds for $m$ 2) What is $m$ if T is also balanced? [with $m^h=l$ for maximum leaf in a m-ary tree $m^4=81$ then m=3 ...
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2answers
142 views

existence of a spanning tree

Let $T$ and $T'$ be two spanning trees of a connected graph $G$. Suppose that an edge $e$ is in $T$ but not in $T'$. Show that there is an edge $e'$ in $T'$, but not in $T$, such that ...
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2answers
179 views

Tree problem about preorder notation

Show that an ordered rooted tree is uniquely determined when a list of vertices generated by a preorder traversal of the tree and the number of children of each vertex are specified.
2
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1answer
147 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
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1answer
503 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
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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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0answers
58 views

How can I prove this property of a $d$-ary tree?

I have the following homework (algorithms lecture): Every $d$-ary tree $G=(V,E)$ contains a vertex $v$ such that the size of the subtree with root $v$ is at least $\frac{1}{d+1} \vert V \vert$ and at ...
0
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1answer
307 views

Help in understanding search of Vantage-Point tree

This is my reference: http://stevehanov.ca/blog/index.php?id=130 A vantage-point tree is a way of organizing a set of points so that finding the n-nearest neighbors is as efficient as possible. It ...
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vote
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!
0
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1answer
78 views

Constructing a tree from disjoint graphs

I will preface my question with the definition of a simple tree that applies to my question: -"A simple tree is an undirected and connected graph with no cycles."- I am having difficulty coming up ...
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3answers
837 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 ...
0
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1answer
82 views

Natural order of rational trees?

What would be a natural order of rational trees? Rational trees arise naturally from free algebras if we view a term as a finite tree. For example the term f(a,g(b,c)) could be viewed as the ...
5
votes
1answer
392 views

Every automorphism of a tree with an odd number of vertices has a fixed point

If $T$ is a tree, and $T$ has an odd number of vertices, then $\forall f$, where $f$ is automorphism $\Rightarrow \exists$ fixed point (vertex). What it means: Formally, an automorphism of a tree $T$ ...
0
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1answer
448 views

How to make a parse tree for the following propositional logic formula?

I have a formula $\neg (( q \rightarrow \neg q) \vee p \vee ( \neg q \rightarrow ( r \wedge p)))$. As it contains 3 subformulas between the $\vee$'s, how can I put it into a parse tree. Would it be ...
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2answers
347 views

Proof for Full Binary Tree Using Handshaking Lemma?

I asked a question a few days ago and figured out the proof for this theorem using induction. ...
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1answer
399 views

Proving terminal vertices and total vertices of a full binary tree?

I am trying to make a proof by induction of the following theorem. ...