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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2
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1answer
38 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 ...
5
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3answers
65 views

Name of the generalization of quadtree and octree?

What is the name of the equivalent of quadtrees and octrees in n-dimension ?
1
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2answers
52 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 ...
1
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2answers
35 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
49 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.
0
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0answers
20 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
votes
1answer
783 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
votes
1answer
104 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 ...
1
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2answers
103 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?
0
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2answers
59 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
13 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?
0
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1answer
62 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 $d_1+\ldots+d_n=2n-2$.
3
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1answer
134 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 ...
0
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2answers
111 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.
0
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1answer
28 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, ...
1
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1answer
235 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 ...
-2
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1answer
212 views

Parse tree parenthesis notation [closed]

Can anyone help me understand this in a tree format? What the indentation means? How do I represent this in a tree? ...
0
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1answer
92 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 ...
1
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0answers
110 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
83 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.
0
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1answer
58 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 ...
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
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1answer
41 views

Recursive trees

Use the method of recursive tree to determine a good asymptotic upper bound (as tight as possible) for the following recurrence and prove your answer using induction (assuming that $T(n)$ is a ...
4
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2answers
291 views

Spanning Trees of the Complete Graph minus an edge

I am studying Problem 43, Chapter 10 from A Walk Through Combinatorics by Miklos Bona, which reads... Let $A$ be the graph obtained from $K_{n}$ by deleting an edge. Find a formula for the number ...
0
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0answers
32 views

Search L leaves smaller than node N

So in my binary $kd$-tree I have a node $N$. Now I search for the number of leafs $L$ "on the left" side of $N$ (this includes the left child branch of $N$ and all parents where the node is a right ...
0
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1answer
50 views

Non-Isomorph trees of a graph

Please consider this graph How many non-Isomorph trees with 4 vertex has this graph? Is there any formula that show number of non-Isomorph trees with $n$ vertices? thanks
0
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1answer
53 views

Depth of BFS Tree With Different Root Nodes

I need to either prove or disprove that for any node of a graph, the depth of the BFS tree using this node as root is always the same. My intuition is that this is true, but I'm having difficulty ...
2
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0answers
42 views

Number of nodes with even offspring

I've been working on a combinatorics assignment, and while the last few questions had clever solutions which didn't involve functional equations and the use LIFT, I fear I'm at my end. Given a ...
1
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1answer
31 views

Does the Prim algorith always create the same tree despite the starting node?

Does the Prim algorith always create the same tree despite the starting node? PD: sorry for my english.
0
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1answer
76 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 ...
0
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0answers
44 views

How to calculate branching factor of uniform tree

For a uniform tree of depth $d$ and if a particular problem has $N$ nodes then the $b*$ branching factor is $N + 1 = 1 + b* + (b*)^2 + ... + (b*)^d$. For a depth of 5 and N = 52 how is it that the ...
0
votes
2answers
48 views

Finding an equation for a growth formula

Given a tree that has three nodes each level I want to find the formula that predicts the number of all nodes with a given tree height. I fitted the data into Numbers with an exponential function ...
0
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0answers
100 views

max number of keys in a 2-3-4 tree

Let $M(L)$ be the largest number of keys (a $2$-node has $1$ key and two children, a $3$-node has $2$ keys and $3$ children, and a $4$-node has $3$ keys and $4$ children) in a $2-3-4$ tree that ...
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 ...
2
votes
1answer
129 views

spanning trees of an edge transitive graph

Let $G$ be an edge transitive graph. Let $t(G)$ be the number f spanning trees on $G$. Show that each edge lies in exactly $\tfrac{(n-1)t(G)}{m}$ spanning trees. Where $|V(G)|=n$ and $|E(G)=m$. ...
2
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2answers
1k views

Can Prims and Kruskals algorithm yield different min spanning tree?

In this problem I am trying to find the min weight using the Prims and Kruskals and list the edges in the order they are chosen. For Prims I am getting order ...
0
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2answers
349 views

creating a Binary tree based on a prefix expression

I want to find the value of a the prefix expression -/+8,10,2*3,2 and build its binary tree I am trying to learn this for a math course, but have absolutely no clue ...
0
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0answers
192 views

D ary tree node math

A d-ary tree is a rooted tree in which each node has at most d children (c) Suppose the tree has n nodes. What is the minimum the depth could possibly be, in terms of n and d? You can leave your ...
2
votes
4answers
861 views

How many labeled trees exist on n vertices with exactly 3 vertices of degree 1?

My combinatorics class is covering spanning trees right now and one of the questions being asked is "What is the number of labeled trees on n vertices with exactly $3$ vertices of degree $1$?" I've ...
0
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1answer
60 views

Counting the number of trees on $[n]$

Let $T_{n}$ be the number of trees on $[n]$. Explain the identity below in terms of $T_{n}$ and prove it. $2(n-1)n^{n-2}=\sum_{k=1}^{n-1}\binom{n}{k}k^{k-1}(n-k)^{n-k-1}.$ So far I've got that ...
0
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1answer
25 views

Question Concerning Family of Trees

I have the following problem where I am asked to construct a family of trees (one for each $n$) that have exactly 2 leaves. I am having difficulty with this problem mainly because I cannot find a ...
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0answers
35 views

automorphism of a rooted tree

Nowadays i'm working with tree automorphisms. I couldn't find information about rooted tree automorphism concerning the root. Does an automorphism of a rooted tree fix the root or not? Logically it ...
1
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2answers
71 views

In a tree, is there always a sink where every longest path ends in?

Let $T$ be an undirected tree. Can we always find a leaf vertex $s$ such that every longest path of $T$ has its other endpoint in $s$? It's easy to see that every longest path passes through the ...
1
vote
2answers
97 views

Some equivalence relation from flipping binary trees

I know almost nothing in combinatorics, so this question might be very easy, or well-known. Fix a number $n$. We will consider rooted planar binary trees with $n$ leaves. We will distinguish between ...
0
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0answers
16 views

Measuring values at nodes of two independent but now connected trees

I am not sure if this the right forum for this question and I hope I am providing enough details on what I want to accomplish. I have an application that has multiple trees - Tree 1 - is categories/ ...
2
votes
1answer
76 views

Given an $n$ level tree with $b$ branches at each node, how many unique paths are there from the root to the leaves?

I have a tree where, at each node, it splits into $b$ branches for a total number of $n$ levels. I enumerate the paths from the root to the leaf nodes. For example, if $n = b = 2$ then I have the ...
1
vote
1answer
38 views

Connection Trees and Partition

In our lecture we just had a short excursion into the tree-world. But the professor mentioned some connection between Ramsey and König's Infinity Lemma (If $T$ is a tree of hight $\omega$ with all ...
3
votes
2answers
178 views

König's Infinity Lemma and Aronszajn Trees

I am working through the notes of my Set Theory lecture. There my professor wrote: 'Is there an uncountable $\kappa$ such that König's Infinity Lemma holds for $\kappa$? There are models where ...
2
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1answer
81 views

questions about binary search tree

Show that every n-node binary search tree is not equally likely (assuming items are inserted in random order), and that balanced trees are more probable than straight-line trees. How is it prove ...