0
votes
1answer
22 views

Generators Trees in a Tree

My question is very short: How many spanning trees have a tree? Thanks in advance
0
votes
0answers
18 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 ...
14
votes
2answers
270 views

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

Post-Bounty Edit: Still no answer; will gladly accept if someone can provide a reference. Earlier Edit: It appears that the answer is "yes," either by an already existent publication or by ...
0
votes
0answers
26 views

prove splits compatible if and only if edge-split

"Prove that if $e_A$ and $e_B$ are distinct edges of a binary $X$-tree $T$ and $C=A\Delta B$(symmetric difference), then the splits $\sigma(A), \sigma(B)$ and $\sigma(C)$ are compatible if and only if ...
0
votes
0answers
16 views

Representing trees in Set builder notation?

Is there a way to represent graphs and minimum spanning trees using set builder notation? e.g. I have a weighted graph of n nodes, all connected to each other in a mesh network manner. I am to ...
0
votes
1answer
59 views

Determining the total degree of a tree

At the start of the solution, I understand that any tree with four vertices has three edges. I don't understand the next statement: "Thus the total degree of a tree with four vertices must be 6." ...
0
votes
1answer
48 views

How do I construct a minimum spanning forest?

I realize that a minimum spanning forest in a weighted graph is a spanning forest with minimal weight. Does this mean that I construct it by turning all of the trees into spanning trees?
1
vote
1answer
54 views

What is the difference between a forest and a spanning forest?

If a graph is labelled as a forest it does not contain any cycles, meaning it consists of all trees, which I realize can even be a single node (since that is technically a tree). If a graph is ...
0
votes
2answers
137 views

How many trees are in the spanning forest of a graph?

Spanning forest is defined by the following definition: A forest that contains every vertex of G such that two vertices are in the same tree of the forest when there is a path in G between these two ...
0
votes
1answer
95 views

Bipartite Graphs and Trees Questions

Which of the claims below is not equivalent to the rest? 1) Every cycle in a graph "B" has an even length 2) Graph "B" is bipartite 3) Graph "B" has two components that are connected. 4) Graph "B" ...
3
votes
2answers
82 views

How many vertices does this tree have?

Suppose that $T$ is a tree. It has $e$ edges and $n$ vertices, and $\overline{T}$ has $10e$ edges. What is n? I think $n = 1$ is a solution, because $T$ can have no edges then, so $0=10*0$. A ...
3
votes
2answers
294 views

Show that if G is a simple graph with at least 4 vertices and 2n-3 edges, it must have two cycles of the same length.

For $n\ge4$, let G be a simple n-vertex graph with at least $2n - 3$ edges. Prove >that G has two cycles of equal length. (West's Introduction to Graph Theory Q 2.1.42) I am trying to prove the ...
0
votes
1answer
48 views

Binary Search Tree Traversals

Draw a BST when you insert, $O,V,E,R,F,L,C,W$ from left-to-right and determine the order of the nodes when using post-order traversal and pre-order traversal. My attempt at drawing the BST: ...
1
vote
1answer
209 views

Graph Theory(trees) problem?

I am practicing for my Discrete Math final and came across this question on trees in my textbook(Rosen). Suppose 1000 people enter a chess tournament. Use a rooted tree model of the tournament to ...
1
vote
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.
1
vote
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) = ...
0
votes
1answer
74 views

Inserting values left to right in a binary search tree

What does it mean to build a binary search tree by inserting values from left to right starting from an empty tree? The "left to right" part confuses me..I know how to build one by normally inserting ...
0
votes
2answers
1k views

How to find non-isomorphic trees?

"Draw all non-isomorphic trees with 5 vertices." I have searched the web and found many examples of the non-isomorphic trees with 5 vertices, but I can't figure out how they have come to their ...
2
votes
1answer
592 views

Number of distinct path in a graph with $n$ vertices

Let $T = (V , E)$ be a tree with $|V | = n\geqslant 2$. How many distinct paths are there (as sub graphs) in $T$? I already have the answer to this question as $(n/2)$. The problem that I'm having is ...
1
vote
2answers
126 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
241 views

No of labeled trees with n nodes such that certain pairs of labels are not adjacent.

What is the number of trees possible with $n$ nodes where the $i$th and $(i+1)$th node are not adjacent to each other for $i \in \left[0,n-1\right)$ and $$i/2 = (i+1)/2.$$ (integer division) (nodes ...
0
votes
0answers
21 views

Simplification of a dervived binary tree with n nodes [duplicate]

hi I need help with this problem how do simplify this equation and what are the steps and approaches to this problem
0
votes
0answers
291 views

What is the algorithm to sort 5 elements in 7 binary comparisons?

I'm tasked with finding the algo that sorts 5 elements in 7 binary comparisons. (The 7 is derived from ceilingFunction(log 5!), which our text states is the minimum number of comparisons required for ...
2
votes
2answers
897 views

Tree pruning question…

all. I'm facing the question: "A chain letter starts when a person sends a letter to five others. Each person who receives the letter either sends it to five other people who have never received it ...
1
vote
1answer
385 views

How can I tell how many non-isomorphic unrooted trees with 6 edges exists without drawing them all?

Typically my professor asks that we draw them all, but I would like to save some time to confirm how many I need.
5
votes
2answers
245 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
182 views

has deleting node in a binary search tree Displacement feature?

I am developing an academic project about graph and tree theory.I searched a lot but I didn't find a clear answer. In a part of project we want to delete some nodes from tree for example we want to ...
2
votes
1answer
265 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 ...
2
votes
1answer
263 views

Proof about trees

Show that in any tree there exists a node such that, if we remove this node and the edges adjacent to it, we will obtain trees which have at most n/2 nodes (the removed node is not counted ...
1
vote
3answers
92 views

Comparison trees

You have 60 coins. You know that 1 coin is either lighter or heavier than the other coins. How many comparisons are needed in a worst case scenario to discover which coin is the false one and ...