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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566 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 ...
6
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2answers
141 views

Graph and in-Degree and Drawing

We have in and out degree of a directed graph G. if G does not includes loop (edge from one vertex to itself) and does not include multiple edge (from each vertex to another vertex at most one ...
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1answer
32 views

Graphs that are almost trees

What's the name of rooted trees in which arbitrary connections between vertices of consecutive levels are allowed? (The level of a vertex is its distance to the root.) I.e.: All parents of a vertex ...
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3answers
94 views

Acyclic graph must have a leaf

It is a theorem that every acyclic graph must have a leaf, ie. A vertex with degree 1 at most. Intuitively, it makes sense as any vertex with more degree would be connected to at least 2 vertices ...
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0answers
53 views

Ordinals as Trees

I'm trying to understand countable ordinals and their tree representation. I understand that $\omega$ is the first "non branching tree" of infinite height. I also understand that the exponent of ...
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2answers
5k views

Determining Ambiguity in Context Free Grammars

What are some common ways to determine if a grammar is ambiguous or not? What are some common attributes that ambiguous grammars have? For example, consider the following Grammar G: $S \rightarrow ...
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1answer
25 views

Height of the tree : $T(n) = 4T(n/4)+2T(5n/8)+T(n/8)+\theta(1)$

Let the tree described by $T(n) = 4T(n/4)+2T(5n/8)+T(n/8)+\theta(1)$ Can someone explains why the height is $\log_{8/5}{n}$ I don't know how to proceed
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1answer
53 views

Infinite Search Tree Probibility

I have a question on Search Trees. I have a balanced, infinite, search tree. If you check a node at level $l$, the probability of finding a solution at that node is $p^l$. Questions The first ...
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1answer
233 views

Prove through structural induction that a binary tree has an odd number of nodes

A full binary tree is a binary tree where every node has either 0 or 2 children. Prove that every non-empty full binary tree has an odd number of nodes. I dont know how to get started with this ...
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1answer
245 views

Graph Min Cut Problem

The idea is to give an Flow Network in which the minimum cut goes through a lot of edges. So adding one unit to each edge will change the min cut. The following figure, as a counter example, shows a ...
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0answers
68 views

Proof by Induction for Splay Tree?

I'm preparing for an exam about Trees. One of the questions that appear in Mark Allen Weiss' "Data Structures and Algorithms Analysis in C++" is: Prove by induction that if all nodes in a splay tree ...
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1answer
64 views

$k$-connected graphs containing trees

I've encountered the following problem in the book "Graphs and Digraphs" and I'm not sure how to do it. Show that every $k$-connected graph contains any tree of order $k+1$ as a subgraph.
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1answer
110 views

Complete Binary Tree [closed]

A balanced binary tree is a full binary tree in which every leaf is either at level l or l-­1 for some positive integer l. The set of balanced binary trees is defined recursively by: Basis step: A ...
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1answer
81 views

Random Binary Search Tree, expected value of nodes with two children

In class, the professor showed that using a uniform random permutation $$ X_1,..., X_n$$ (each being i.i.d.) we can construct a Binary Search Tree by inserting the values in to the tree by their ...
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1answer
47 views

Calculating the average degree/valency of vertices

If I were to let T be a tree with n vertices, what would be the average degree/valency of the vertices in T? How would I go about calculating this?
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1answer
208 views

Proving that the height of a 2-3 tree is between $\log_3 N$ and $\lg N$

I am stuck on the problem of trying to prove the upper and lower bounds of a 2-3 tree. I think the most natural recourse is to use induction. However, my instructor told me that this was unnecessary ...
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1answer
93 views

Finding a minimum weight spanning tree? [duplicate]

Letting W be the weighted graph created by taking a complete graph K5 on five vertices 1, 2, 3, 4, 5 with the weight of each edge {x,y} given by ({x,y})=x+y, How would I find a minimum weight ...
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0answers
29 views

Proof for the number of leaves for any Binary Search Tree

A property for binary trees is that the number of leaves is the number of full nodes plus 1, in other words, $L = F + 1$ where $L$ is the number of leaves and $F$ is the number of full nodes. What ...
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1answer
66 views

How would I find a minimum weight spanning tree for W?

If I were to let $W$ be the weighted graph formed by taking a complete graph $K_5$ on five vertices 1, 2, 3, 4, 5 with the weight of each edge $\{x,y\}$ given by $(\{x,y\}) = x + y$, how would I find ...
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1answer
227 views

Question about trees, Let T be a tree with n vertices

Are my answers correct to these 3 questions? Let T be a tree with n vertices. 1) What is the average degree/valency of the vertices in T? Average Degree of of ...
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1answer
141 views

How many spanning trees does the cycle graph C2014 have?

How many spanning trees does the cycle graph $C_{2014}$ have? How do I create a bipartite graph and use it to solve this problem?
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1answer
133 views

Postorder and Preorder traversal on a Binary tree

For the tree below, list the labels of the nodes of the tree according to the pre-ordering algorithm, and then re-list them according to the post-ordering algorithm. ...
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1answer
43 views

Graph theory: tree vertices

How can I calculate the number of vertices of a tree knowing he has 33 vertices of degree 1, 25 vertices of degree 2, 15 vertices of degree 3 and all other vertices of grade 4?
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1answer
49 views

Spanning tree with unique paths.

Let $G$ be a connected graph and let $r∈V(G)$. Prove that $G$ has a spanning tree $T$ such that for every edge of $G$ with ends $u$ and $v$, either $u$ belongs to the unique path in $T$ with ends $v$ ...
6
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1answer
1k views

Proving every tree has at most one perfect matching

In trying to prove that every tree, T, has at most one perfect matching, I came across this idea: ...
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1answer
41 views

Expected number of subtree removal in a tree.

I was solving this problem. In a gist the problem is as follows: You are given a rooted tree. On each step you choose a node randomly and remove the subtree rooted by that node and the node ...
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1answer
64 views

Normalization of data in decision tree

After reading through a few references, I have come to know that for machine learning in general, it is necessary to normalize features so that no features are arbitrarily large ($centering$) and all ...
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1answer
53 views

The intersection of $k$ subtrees of a tree $T$ is nonempty.

Let $T_1$, $T_2$, . . . , $T_k$ be subtrees of a tree such that any two of them have a vertex in common. Prove that they all have a vertex in common. Any hints/solutions are greatly appreciated. I am ...
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1answer
34 views

Find the node that we will reach with a given path on a graph (complete binary tree)

This question is regarding a special case of graph i.e. complete binary tree Consider the following tree :- ...
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1answer
67 views

Preorder Traversal

For Each Preorder Traversal, we have multiple Inorder Traversal. this is True or False Conclusion? every one would help me and add some detail.
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1answer
169 views

Visiting Node in BFS and DFS in the same order [closed]

if G be a connected, undirected graph and has at least 3 vertex. we know the order of visiting node from a given vertex in BFS and DFS is the same. which of the following is false? a) G can be a ...
2
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1answer
67 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
43 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 ...
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0answers
31 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 ...
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1answer
47 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 ...
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1answer
71 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 ...
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2answers
733 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 ...
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1answer
151 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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1answer
78 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 ...
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2answers
368 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. ...
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0answers
50 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
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1answer
115 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
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1answer
96 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 ...
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1answer
133 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 ...
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3answers
81 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 ...
2
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1answer
112 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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0answers
38 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. ...
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1answer
57 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
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1answer
26 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.
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2answers
108 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.