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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3
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1answer
20 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.
1
vote
1answer
19 views

A graph with minimum degree $k+2$ contains any $(k+3)$-vertex tree as a subgraph?

Let $k$ be a positive integer and let $T$ be a tree of order $k+3$. If $G$ is a graph with minimum degree at least $k+2$, prove that $G$ contains a subgraph isomorphic to $T$. Any solutions or hints ...
0
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0answers
18 views

Pascal's Identity and Trees

Pascal's Identity states that $n \choose k$ = $n-1 \choose k-1$ + $n-1 \choose k$ since if one element is separated from the rest we can claim that either it is chosen (resulting in $k-1$ elements ...
3
votes
1answer
25 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 ...
1
vote
1answer
57 views

Iterations of Pascal's Identity

Pascal's Identity states that $n \choose k$ = $n-1 \choose k-1$ + $n-1 \choose k$ since if one element is separated from the rest we can claim that either it is chosen (resulting in $k-1$ elements ...
7
votes
2answers
435 views

Is the graceful labeling conjecture still unsolved?

From the Wikipedia article on graceful labeling: ... A major unproven conjecture in graph theory is the Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, which hypothesizes that ...
3
votes
1answer
116 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 ...
0
votes
1answer
26 views

How to understand the perfect binary tree formula?

I got this paragraph by reading "python algorithm", in which it mentioned `some knights participate in an knockout match, how many mathes do they need to produce the winner. It's answer says: I'm ...
14
votes
2answers
290 views

Does every “balloon” (dragon, tadpole, canoe paddle) admit a graceful labeling?

8/18/14 Edit: If anyone has a copy of a related reference, then I would be happy to see it. For now, I am accepting the answer below and considering the question answered in the affirmative: Yes. ...
3
votes
0answers
35 views

What are the algebraic and topological properties of a tree (graph theory), and what are any possible connections?

I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup ...
2
votes
1answer
94 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 ...
2
votes
1answer
50 views

Infinite Tree Probability Question

Suppose I have 10 dollars and I'm able to make fair 50/50 bets like flipping a coin. Now suppose each bet is for 1 dollar. What is the probability that if I keep making bets until I hit 0 dollars ...
-1
votes
1answer
47 views

The union of two connected graphs is connected [closed]

Let $G = (V,E)$ be a graph and let $H_1 = (V_1,E_1)$ and $H_2 = (V_2,E_2)$ be two connected subgraphs of $G$ that have at least one node in common. Prove that the graph $H = H_1\cup H_2 = (V_1\cup ...
0
votes
3answers
37 views

How to write a summation function that counts the number of nodes in a tree?

I come from a programming background and am interested in learning how to represent some things as simple equations, as an entry into thinking mathematically. How do you represent a tree structure as ...
-1
votes
0answers
18 views

Tree stored as adjacency matrix

Can tree be stored as an adjacency matrix? My answer: No. Reason: all though it can be stored as adjacency matrix, but the hierarchy is not maintained when stored as a matrix. Also, who are the ...
0
votes
1answer
37 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. ...
0
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0answers
13 views

Tree of arity n: How to call a vertex that has only k (k<n) children?

What is the correct adjective for a vertex in an n-ary tree that has only k children (k < n)? I was thinking of something like "unsaturated", but I don't know if that is the correct word for this. ...
2
votes
1answer
272 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 ...
0
votes
1answer
29 views

Number of distinct Binary tree formed with respect to height h

How many types of distinct Binary Tree can be formed with a height of h? assuming height starts from 0 when the tree has only the root. example: if the height of tree is 1 then root-leftchild ...
0
votes
1answer
20 views

Proof about spanning tress in graphs

Let $G=(V,E)$ be a graph and $T_i=(V,F_i),i=1,2$ two disjoint spanning trees in $G$. Let $f_1 \in F_1$. Prove that there is $f_2\in F_2 $ such that $T:=T_1-f_1+f_2$ is a spanning tree.
3
votes
2answers
2k 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 ...
0
votes
0answers
21 views

How does inserting N objects one at a time into an ordered AVL tree yield an efficient sorting algorithim

If we assume reblalancing an AVL tree of height n after an insertion or deletion takes O(n) operations. How does inserting N objects one at a time into an ordered AVL tree yield an efficient sorting ...
0
votes
2answers
44 views

Prove that if G is a tree in which all vertices have odd degree then G has odd size.

Prove that if G is a tree in which all vertices have odd degree then G has odd size. Good night, do not know how to approach this "prove". Can you give me tips to solve it?. Please.
0
votes
1answer
21 views

Formula for number of “root” nodes in a tree where Parent shares child nodes?

If I have a tree like this: {a},{b,c},{d,e,f},{g,h,i,j} in this case we have a total of 10 nodes. Is there any equation where given "10" I can calculate how many bottom nodes there are (answer: "4" ...
1
vote
3answers
73 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 ...
0
votes
1answer
34 views

How to compute a marginal probability

Given a weighted graph, using the Kirchhoff's matrix tree theorem, how can I compute the marginal edge presence probability: $$P_\beta(ij)=Z_\beta^{-1}\sum_{\text{T spanning tree:$(i,j)\in ...
0
votes
2answers
50 views

Why is the height of a heap defined as $\lg n$?

I'm a bit confused about why the height of a heap (or a binary tree in general) is given by the floor of $\lg n$. E.g. if you have a tree with 7 nodes, you would get $h = 0$ instead of $h = 2$. Isn't ...
0
votes
0answers
38 views

Min. Spanning Tree - Same weight

Prove that every minimum spanning tree of a connected graph, $G$, has the same maximum edge. Intuitively, this makes sense to me. You need to have that heavy edge because that is the cheapest ...
0
votes
2answers
47 views

Tree with $k$ edges is a subgraph of any graph with all vertices of degree $\geq k$.

Let $T$ be a tree with $k$ edges. Let $G$ be a graph where every vertex has a degree of at least $k$. Show that $T$ is a subgraph $G$. I know this implies that in a graph where every vertex is at ...
2
votes
1answer
51 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)?
0
votes
1answer
24 views

Generators Trees in a Tree

My question is very short: How many spanning trees have a tree? Thanks in advance
0
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2answers
31 views

Number of spanning trees of a graph (behind the formula)

Given $G$ a subgraph of $K_n$ s.t. $G$ has $n$ vertices with adjacency matrix $A$; why is $$\sum_{T \text{ spanning tree of }K_n}\prod_{(i,j)\in T}A_{i,j}$$ the number of spanning trees? I can't get ...
1
vote
1answer
126 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 ...
3
votes
0answers
39 views

Proof on Dyadic Trees [Smullyan: First-Order Logic, chapter 1, section 0]

I'm having difficult with a proof from Smullyan's First-Order Logic, Chapter 1 Section 0 (Reprint, Dover 1968, p. 4): Prove: In a dyadic tree, define x to be to the left of y if there is a ...
3
votes
2answers
114 views

Number of all labeled, unordered rooted trees with $n$ vertices and $k$ leaves.

I've been trying to do the following exercise: The problem Find the number of all labeled, unordered rooted trees with $n$ vertices and $k$ leaves. I know that I should try to write an ...
1
vote
0answers
31 views

Find tree diameter or center

I want to find center in a graph that doesn't have cycles. I heard, that this is how I find a diameter: Take random vertex A Find such vertex B, that distance to it is maximal Find such vertex C, ...
0
votes
0answers
27 views

Can a non-cyclic infinite proof tree with always-reachable provable nodes be used to construct a proof?

Suppose that I have a finite number of basic elements x,y,z ... and a finite number of operators +, * ... Terms X,Y,Z ... are created by combining basic elements and operators. For example, x+y, and ...
0
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0answers
19 views

$k$ edge-disjoint $r$-arborescences in an acylic digraph

An $r$-arborescence of a digraph $D$ is a rooted spanning tree with root $r\in V(D)$ in which all the edges of $D$ are directed away from $r$. I would like to prove the following: I have thought ...
2
votes
1answer
33 views

tree structure on classes of elements in GL_2 over a field with discrete valuation

this is my first question here, so I hope I am doing it right. :) I'm currently reading a paper about the tree of GL_2 over a discretely valued field (similarly to Serre). Here's the setting: $k$ an ...
8
votes
0answers
111 views

Free medial magmas

A medial magma is a set $M$ with a binary relation $*$ satisfying $(a*b)*(c*d) = (a*c)*(b*d)$. Medial magmas constitute an algebraic category $\mathsf{Med}$, therefore there is a functor $\mathsf{Set} ...
2
votes
1answer
56 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 ...
4
votes
3answers
317 views

Counting $k$-ary labelled trees

The (full) binary counting tree problems gives the number of binary trees can be formed using $N$ nodes $T(n)= C_n$, where $C_i$ are the Catalan numbers. The recursion form is $T_n = ...
3
votes
0answers
41 views

maximizing the inverse degree in a graph

The inverse degree in the graph $G$ is defined as \begin{align*} r(G) = \sum_{i=1}^N \frac{1}{d_i}, \end{align*} where $d_i$ is the degree of node (vertex) $i$. Is the connected graph with maximum ...
4
votes
2answers
209 views

On the number of caterpillars

A caterpillar is a tree with the property that if all the leafs are removed then what remains is a path. Could you help me to prove that there are $2^{n-4}+2^{\lfloor n/2\rfloor-2}$ caterpillar on $n$ ...
5
votes
2answers
146 views

An identity on the number of trees

Let $T_n$ be the number of labelled trees on $n$ vertices, then $$ T_n=\sum_kk\binom{n-2}{k-1}T_kT_{n-k} \tag{1}$$ Using this question, I was able to prove that $$ T_n= \frac{n}{2} \ ...
1
vote
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 ...
0
votes
2answers
24 views

Determine the minimum number of weighings to find the counterfeit coin.

Here's the full problem: We have 20 coins, 1 of which is counterfeit (too light). Determine the minimum number of weighings to find the counterfeit coin. Okay so is used the formula $$h=\left ...
0
votes
1answer
24 views

Finding the number of spanning trees on a set of vertices.

I need to find the number of spanning trees on $V = \{1,2,3,4,5,6,7,8,9\}$, where $\{1,2,3,4\}$ are leaves. Can anyone tell me how?
1
vote
3answers
82 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
169 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 ...