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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2answers
34 views

Find all possible topological-sortings of graph G

A topological ordering of G is an ordering of the nodes as $v_1,v_2,...,v_n$ so that all edges point "forward": for every edge $(v_i,v_j)$, we have $i<j$. Moreover, the first node in a ...
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1answer
38 views

How can I draw a tree to represent combinations?

I understand how to systematically draw a tree for permutations. How do you do this for combinations? In my book, I don't see a system to avoid repetitions. I'd like to draw a tree of 5C3 if possible. ...
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2answers
98 views

Trees with no vertex of degree 2 have more leaves than internal nodes

There is a question asked by portal about Tree having no vertex of degree 2 has more leaves than internal nodes so we want to prove this claim by induction and an answer from Micheal Biro suggested ...
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1answer
83 views

What is a “linear chain” in Graph Theory?

What is a linear chain in the context of graphs and trees? For example: a topological sort forms a linear chain What does a linear chain mean in the example above? Another example from ...
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1answer
37 views

Circuits and Trees

Given a graph G, can it be split into 2 sets of graphs($ G_1, \; G_2 $) such that, $G_1$ consists only trees and $G_2$ consists only circuits ? In other words: Is it possible to construct any graph ...
3
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1answer
253 views

Tree having no vertex of degree 2 has more leaves than internal nodes

If $T$ is a tree having no vertex of degree 2, then $T$ has more leaves than internal nodes. Prove this claim by a) induction, b) by considering the average degree and using the handshaking lemma. ...
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1answer
61 views

Polynomial time algorithm for finding the chromatic sum of a tree.

As the title goes, a polynomial time algorithm for finding the chromatic sum of a tree is required. NOTE: Finding the chromatic sum of a graph is also called the sum coloring problem - The sum ...
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1answer
28 views

Intersection of all possible spanning trees of a connected, simple graph.

Is the intersection of all possible spanning trees of a simple, connected graph $G$ equal to the graph $(V_{G}, \varnothing)$? I'm not sure if this is a trivial question or not. Although I'm going to ...
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0answers
69 views

Why is the intersection of countably many homogeneously Suslin subsets of $\,^{\omega} \omega$ homogeneously Suslin?

I found this assertion in these notes: The derived model theorem (Steel) right in the beginning on page 3, together with the remark that this is 'not too hard to show'. Unfortunately, I'm ...
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1answer
120 views

Tree with no nodes of degree 2: prove that # leaves ># internal nodes using average degree and handshake lemma

Im really struggling to formalise my thoughts on this one. Basically I understand that if we would allow nodes with degree 2, then we could chain together infinitely many nodes to always produce ...
2
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1answer
26 views

Different trees of weighted graph , please check whether my explanation is correct $?$

Let G=(V, E) be a graph. Define $\xi(G) = \sum\limits_d i_d*d$, where $i_d$ is the number of vertices of degree $d$ in G. If S and T are two different trees with $\xi(S) = \xi(T)$, then ...
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0answers
33 views

Complexity analysis of alpha beta pruning of a full tree

I am trying to understand the derivation of a time complexity for an alpha-beta pruning algorithm but up till now have not found any reasonable recourse. Many recourses claim that if you take a full ...
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0answers
21 views

Yet another curious convolution

Some time ago, I found the following algorithmic problema: Count the number of distinct unrooted, unordered, labeled trees of $n$ nodes where each node has at most $k$ neighbors. Given that the ...
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0answers
82 views

For a k-ary tree with height h, the upper bound for the maximum number of leaves is $k^h$

i want to prove For a k-ary tree with height h, the upper bound for the maximum number of leaves is $k^h$. (assume the complete k-ary tree is a tree that is complete in all levels including the ...
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0answers
20 views

directed trees versus singly-connected directed graphs

According to wikipedia, a singly connected directed graph (a.k.a. Polytree) is a Directed Acyclic Graph whose underling undirected graph is a tree. How is this different from directed trees (trees ...
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0answers
41 views

Is there a name for this particular kind of tree graph?

I've recently encountered a problem which heavily involves analysis of structures analogous to weighted trees with no nodes of degree two (such a node along with its adjacent edges would be ...
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0answers
32 views

Binary Minimum Spanning Tree (from complete graph)

Given a weighted complete graph (or more exactly, a matrix of pairwise metric distances between vertices), I need to find a good approximation of the binary spanning tree of lowest total cost. There ...
3
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2answers
183 views

Graph Theory: Trees, leaves and cycles

So, a vertex is called a leaf if it connected to only one edge. a) Show that a tree with at least one edge has at least 2 leaves. b) Assume that G = (V, E) is a graph, V ≠ Ø, where every vertex ...
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1answer
78 views

Counting spanning trees and hamiltonian paths

Assumption : For any connected graph, every hamiltonian path is a spanning tree but not the other way around. If the assumption is wrong there is no need of reading any furhter. So is the assumption ...
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0answers
53 views

Extract the overall “structure” (“backbone”) of a set of vertices…

My main problem is that I am struggling to find a good graph-theoretical formulation of my problem, let alone a formal name for it (if it already exists, as I suspect it should)… Informally, given a ...
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0answers
21 views

Name of operation: changing the root of a rooted tree

What is (if any) the name of the operation of changing the root of a rooted tree? Picking a vertex which is not the root, then reorienting the edges in such a way that the vertex becomes the root?
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0answers
26 views

is the Root of a binary Tree counted as a node

I am working on this Homework questions and there's one thing I can't seem to understand. We are trying to proof using structural induction that some elements in T hold for the following statement ...
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1answer
44 views

How would one go about solving these types of problems?

I'm totally lost. All I know is it has to do with binary trees and may need to be solved using induction. Show that every 2-tree with $n$ internal nodes has $n + 1$ external nodes. Show that the ...
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0answers
19 views

Questions about this type of problem

Problem Consider the general chip-and-be-conquered recurrence relation: $T(n) = b_1T(n - 1) + b_2T(n - 2) + ... + b_kT(n - k) + f(n)$; for $n >= k$ for some constant $k >= 2$. The ...
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0answers
13 views

Terminology for excluded nodes

Given the following tree: A / \ / \ B C / \ \ / \ \ D E F / \ / \ G H Regarding node ...
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0answers
40 views

Is the following statement about tree true?

For a rooted tree T of oder n, what is the probability of that T contains a balanced tree of order 7? For example, there are total of 719 rooted trees of order 10, IF among all those 719 rooted trees, ...
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0answers
30 views

Defining a Nested Tree (set-theoretic)

I am new to the world of trees and I am trying to make a painless addition to this general definition: Let $X$ be a topological space and $\mathfrak{T}$ be a collection of sets. ...
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0answers
29 views

Perron vector of the distance matrix of a tree

Increasing properties of perron vector of distance matrix from the vertex corresponding to which row sum is minimum
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0answers
21 views

What is considered a unique homeomorphically irreducible tree?

Take, for example, this image that shows the possible trees of size 11 and this tree I created. Based on the 14 trees in the first image, how can I tell if mine is unique or not? Based on the shape ...
1
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1answer
61 views

Construction of rooted tree , please check whether my solution is correct?

Problem is A rooted tree with 12 nodes has its nodes numbered 1 to 12 in pre-order. When the tree is traversed in post-order, the nodes are visited in the order 3, 5, 4, 2, 7, 8, 6, 10, 11, 12, 9, ...
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0answers
30 views

Any implementation of the Roskind-Tarjan algorithm for finding the maximum number of edge-disjoint spanning trees over a graph?

I am looking for an implementation of the Roskind-Tarjan algorithm [1] for finding the maximum number of edge-disjoint spanning trees over a graph. A Matlab implementation would be great. ...
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3answers
40 views

Probability tree diagrams

Exercice : True or false Let $E$ and $F$ be two events of an experiment. $$\mathbb{P}(E| \bar{F})=1-\mathbb{P}(E| F)$$ Solution : Flase due to the Law of total probability we've: ...
3
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1answer
107 views

Number of spanning trees in a complete split graph

A graph is a complete split graph if we can partition it into an independent vertex set and a clique, such that every vertex of the independent vertex set is adjacent to every vertex in the clique. ...
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1answer
94 views

Find a recursive definition for inorder: binary Tree(T) → list(T ) where inorder(T ) is the list of nodes from an inorder traversal of T .

Find a recursive definition for inorder: binary Tree(T) → list(T ) where inorder(T ) is the list of nodes from an inorder traversal of T . I have no idea what this question is even asking me. What ...
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1answer
66 views

How to prove that at Complete Binary Tree (CBT) at height $h$ we have $2^h$ leaves

I try to prove it by induction, please tell me if I'm right... The induction assumption - For every CBT at height $h$ there is $2^h$ leaves. The base of the induction is right (I'm writing this proof ...
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3answers
1k views

Number of spanning trees in a ladder graph

Let $L_n$ be the ladder graph formed from two $n$-vertex paths by joining corresponding vertices. For example $L_4$ is the following I have to find a recurrence $\langle t\rangle$ where $t_n$ is ...
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2answers
129 views

Sum of roots of binary search trees of height $\le H$ with $N$ nodes

Consider all Binary Search Trees of height $\le H$ that can be created using the first $N$ natural numbers. Find the sum of the roots of those Binary Search Trees. For example, for $N$ = 3, $H$ = 3: ...
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1answer
59 views

Dijkstra's algorithm, am I or the teacher mistaken?

Imagine that Dijkstra’s algorithm has been used to show the length of the shortest path from $a$ to $g$ in the graph in figure 1. Which of the following vertices is added first to the set $S$? It's ...
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1answer
70 views

Dijkstra’s algorithm / path is this done correctly?

im doing this assignment and it seems as if my teacher has made a mistake. according to me in order to find the minimum spanning treee from a-z , you start from a and then go to : a,f,d,c,b,e,z,g ...
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0answers
25 views

How many nodes in a K-ary tree with L leaf nodes

Assuming that we have a k-ary tree with L leaf nodes, can the average number of nodes in the tree be calculated if we were to know the average number of children for each node? If not, what other ...
0
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1answer
28 views

Proving that the number of appearances of $i$ in $F(T)$ (Prufer sequence) is $d_i-1$

Let $T$ be a labeled tree on the set of vertices $\{1,...,n\}$, and its sequence of degrees is $d_1,...,d_n$. Prove that for all $1\le i \le n$ the number of appearances in $F(T)$ (Prufer sequence) ...
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0answers
66 views

minimum number of leaves in a perfect binary tree

I'm trying to prove that the number of leaves in a perfect binary tree is at least H+1 where H is the height of the tree. This is what I've done up til now: No of leaves at height $H = 2^H$ Base ...
0
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1answer
50 views

Binary tree node value by level

How can I calculate the value of given node level, for example: (let's use this image I found on Google Images and invert the level: starting at bottom 0..1..2..3..4) Knowing that each node pays ...
4
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2answers
2k views

Show that a graph has a unique MST if all edges have distinct weights [duplicate]

Prove that if all edge-costs are different, then there is only one cheapest tree (minimum spanning tree or MST). (Use contradiction and make sure to keep track of the costs of the different trees ...
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1answer
47 views

$ G=(V,E_1 \cup E_2) $ is a triangle free graph, where $ G_1=(V,E_1) $ is planar and $ G_2 = (V, E_2)$ is a tree. Prove that: $ \chi (G) < 7 $

can anyone help with this, any direction could be helpfull? I've tried using the fact that $ G_1 $ satisfies that it's planar and is triangle free because G is. So we should have $|E_1| \leq 2|V|-4 $ ...
2
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2answers
392 views

Two disjoint spanning trees, spanning subgraph with all even degrees

Show that if a graph has two edge-disjoint spanning trees then it has a connected, spanning subgraph with all degrees even. I start by looking at the union of the two spanning trees. I know it has ...
2
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0answers
74 views

Generating all coprime pairs

The Wikipedia article on coprime integers has a brief section on generating all coprime pairs. All pairs of positive coprime numbers $(m,n)$ (with $m>n$) can be arranged in two disjoint ...
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1answer
57 views

A tree has a root, leaves, and what?

The root of a tree is special, in that it has no parents. The leaves are special in that they have no children. The other nodes each have exactly one parent and more than zero children. Is there a ...
2
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2answers
282 views

How to find the number of all the possible ordered trees with n edges and k leaves?

We know that a tree with n edges have n+1 nodes.So if $|B_{n+1}|$ is the number of all possible ordered trees with n+1 nodes then its true that $C_{n+1} = |B_{n+1}|$ where $C$ is the Catalan ...
6
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1answer
67 views

Let $T$ be the set of full binary trees. In what way $T^7 \cong T$?

I was reading the slides of a talk by Tom Leinster. I have trouble understanding the last line of page 17 (pages 1-15 are irrelevant and can be skipped). Could someone please explain it to me? If I ...