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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35 views

find an algorithm to find MST in linear time while each edge has the same weight

I have been disscussing this problem with a lot of my friends . However no solution has been found. let G= w is a weight function for each e in E w(e)=1 find MST of G in O(|V|+|E|) thanks
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1answer
73 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 ...
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1answer
457 views

Minimum number of nodes in balanced binary search tree

I'd like to know if anyone could help me verify a recursive formula for the minimum possible number of nodes a binary search tree would require to be balanced. So far, I know that the recursive ...
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1answer
137 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" ...
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0answers
97 views

Structural induction on internal nodes of a binary tree

I know my language is not super mathematic, but I want to make sure I have the logic down. Here is my proof for the number of internal nodes in a binary tree being equal to the floor_function(n/2), ...
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1answer
37 views

How I can prove the one order homology of a tree is zero

When T is a tree and d1 is boundary operator fromC_1(T) to C_0(T) how to prove kernel of d1 is {0} I think acyclic is key point but i don't know next step.
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2answers
92 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 ...
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1answer
68 views

Parse Trees - Arithmetic Expressions

In regards to the right side of this expression (c * (a-b)) how is it factored to include (-) instead of * and then (-) again? I cant understand what steps my teacher made to do this.
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2answers
375 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 ...
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1answer
86 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: ...
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1answer
66 views

Infinite sequence of trees that are not subgraphs to each other

This is from a set of exercises and I am stuck to this. Please, have in mind, that I want to understand how it's solved, I am not just looking for a solution. Define an infinite sequence of trees ...
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0answers
39 views

Can an arbitrary network of nodes be effectively visualized as a circular “treemap”?

We all know that a treemap is effective for visualizing hierarchical tree data (i.e. where there are only 1 to many relationships like in a computer file system): But how difficult algorithmically ...
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2answers
95 views

A tree $T$ with 50 end-vertices has an equal number of vertices of degree 2, 3, 4, and 5 but contains no vertices of degree greater than 5.

What should be the order of $T$? So I know a graph $G$ is a tree if every two vertices of G are connected by a unique path
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1answer
53 views

What's an efficient algorithm for walking to a minimum spanning tree?

Given a connected directed acyclic graph $G(V, E)$, is there an algorithm for changing a spanning tree to a minimum spanning tree through a series of edge swaps? We can use Prim's or Kruskal's ...
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1answer
38 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 $\chi(G) \le 4$ ...
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1answer
85 views

Finding graphs with a given number of spanning trees

All of the graphs considered in this question are connected. We can find the number of spanning trees $t(G)$ of $G$ using Kirchhoff's matrix-tree theorem or the deletion-contraction method. I'm ...
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1answer
106 views

Find Minimal Spanning Tree Using Prim's Algorith

What will be the minimal spanning tree using Prim's Algorithm for this graph Also can i draw a tree and assign the weights as i like,will there be a minimal spanning tree for such a graph
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1answer
49 views

Some questions about first-order logic (arising from a book by Raymond M Smullyan)

Recently, I got confused when reading a book about first order logic written by Raymond M smullyan. Chapter 1 page 9:When introducing the notion "Formation tree", smullyan define a formation tree for ...
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0answers
58 views

Proof of existing path on Depth-First-Search spanning tree

Let $G$ be an undirected connected graph, and $T$ the directed spanning tree of $G$, which I got by performing a DFS on $G$. If $H$ is a complete subgraph of $G$, how can I proof that there a path in ...
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2answers
45 views

Proof that spanning trees can be converted into one another

Let $T_1$ and $T_2$ be two spanning trees of a graph $G$. Prove that if $e$ is an edge of $T_1$, there exists an edge $f$ in $T_2$ so $T_1-\{e\}+\{f\}$ also is a spanning tree. For $\{e\}\subset ...
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1answer
44 views

Finding the parent of a node in recombining binomial tree

I have posted an earlier question: Finding the child node in the recombining binomial tree. Now I would like to find the parent of a node in recombining tree. The tree looks like this: Now I need ...
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0answers
282 views

Finding a spanning tree using exactly k red edges in a graph with edges colored by red/blue in linear time.

So we have a graph $G$ with its edges colored by red and blue. we are asked to find a deterministic linear time algorithm that given a parameter $K$ finds a spanning tree of G using exactly $K$ red ...
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1answer
71 views

The complement of spanning trees is covered by a union of cycles

Is it true that in any (connected) graph $G=(V,E)$, if $T$ is a spanning tree than its complement (edge-wise) may be covered by a union of disjoint cycles? Here's a non-complete attempt to prove this ...
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1answer
47 views

Proofs with binary trees [duplicate]

Now I have a binary tree which is How would I go about proving binary tree with $n$ leaves has exactly $2 n - 1$ nodes ?
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1answer
507 views

Number of nodes in binary tree given number of leaves

How would I prove that any binary tree that has n leaves has precisely $2n-1$ nodes ? Given that a binary tree is either a single node "o" or a node with the left and right subtrees contains a binary ...
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1answer
53 views

Variance of Height of Tree

What is the asymptotic variance of the height of rooted plane trees (ie rooted, unlabelled, ordered trees with unbounded node degree) and of ordered binary trees (ie rooted, unlabelled, plane trees, ...
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0answers
36 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 $T$?
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1answer
79 views

Are the structure of logical expression based on formative constructions like sequences or trees ?

Recently, I get confused when reading the book Principles of Mathematical Logic written by D. Hilbert. How to define the term 'logical expression'? I just envisage that it might be defined as anyone ...
2
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1answer
40 views

Number of trees of a certain size

Given a branching factor $b$ and a tree height $h$, a complete tree has $\sum_{i=0}^h b^i$ nodes. Define a partial tree as a sub-tree of the complete tree, with the same root. How many such partial ...
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3answers
73 views

Name of the generalization of quadtree and octree?

What is the name of the equivalent of quadtrees and octrees in n-dimension ?
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2answers
59 views

Construction Types or Type Constructions?

In any (simple) type theory there are base types (i.e. the type of individuals and the type of propositions) and type builders (i.e. $\rightarrow$, which takes two types $t,t'$ and yields the type of ...
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2answers
42 views

How to go from Tree to Total orders

Given a tree $T=(X,E)$, is it guaranteed for any orientation of the edges $E$, there exist a strict total order preserves it? For instance, let $X=\{x_1,x_2,..x_n\}$ and $E=(x_i,x_{i+1})$ the result ...
2
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1answer
50 views

Counting problem (should use Cayley's formula)

How many trees above $V=\{1,2,3,4,5,6,7,8,9\}$ are there, such that $deg(4)=5$? I know I should use Cayley's formula somehow.
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0answers
21 views

Can a search-tree be reversed in order to find more solutions?

consider a search-tree that we already know that if we start from the start-node (S), and go to nodes A1, B2, C1, then we get to the Goal-node. We have this solution, but we don't know the other nodes ...
2
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1answer
2k views

How to draw all nonisomorphic trees with n vertices?

I have a textbook solution with little to no explanation (this is with n = 5): Could anyone explain how to "think" when solving this kind of a problem? (for example, drawing all non isomorphic ...
3
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1answer
117 views

A Graph as a Union of K forests.

I want to show that a graph G that is a union of k forests has a chromatic number of at most 2k. I have narrowed my problem down to trying to show that any graph G that is a union of n trees has a ...
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2answers
106 views

syntax tree for the word (())()()

I have to create the syntax tree for the word (())()() . That's what I have tried: Could you tell me if it is right?
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2answers
66 views

Determining number of parent node on an n-tree.

I'm sorry if this is the wrong one, was unsure if this was computer science, programming, or mathematics related. I'm going with mathematics because it is semi-graph theory related. I have a tree ...
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0answers
15 views

For which graphs do depth first and breadth first produce identical spanning trees?

Is this possible?If yes, what are the conditions it should meet?
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1answer
82 views

How to prove this necessary and sufficient condition for tree in graph theory?

Let $0<d_1\leq\ldots\leq d_n$ be integers. Show that there exists a tree with degrees $d_1,\ldots,d_n$ if and only if $d_1+\ldots+d_n=2n-2$.
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1answer
167 views

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non-trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as ...
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2answers
137 views

Use strong induction to prove number of vertices on complete tree is $2l-1$

Can someone help me construct this proof using strong induction? Use strong induction on $l$ to show that for all $l \geq 1$, a full binary tree with $l$ leaves has $2l-1$ vertices total.
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1answer
37 views

Transforming spanning trees through a sequence of intermediate trees

the problem is as follows: Let $G$ be a connected graph, and let $T_1$ and $T_2$ be two of its spanning trees. Prove that $T_1$ can be transformed into $T_2$ through a sequence of intermediate trees, ...
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2answers
316 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 ...
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1answer
267 views

How to convert parentheses notation for trees into an actual tree drawing? [closed]

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 ...
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1answer
102 views

proof: center of a tree lies on the longest path

how can I make a proof of this property? I mean, given a weighted tree(with positive costs), how can I proof that the center of such a tree lies on the longest path?? I read to the first answer of ...
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0answers
134 views

Software/Applet to Draw Tree Diagrams (for Enumeration Problems)

I need a software/applet/flash file which easily draws tree diagrams for simple enumeration problems: I want to give number of the vertices in each layer, and it draws the diagram which shows all the ...
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2answers
95 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.
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1answer
61 views

Finding missing two edges in a MST in O(m) time

I need to write an algorithm in O(m) time to find the missing two edges of a minimum spanning tree. I am given a graph G(V,E) where m = |E| and n = |V| as an adjacency list, and T, a subset of G, with ...
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1answer
38 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) = ...