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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Number of nodes satisfying a certain property on a binary tree

Fix a large integer $M$ and construct a binary tree as follows. Assign the root node by the integer $0$. If a node is assigned the integer $n$ and $n \leq M - 2$, then $n$ has two children and ...
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1answer
10 views

Easy References for understanding Grossmann and Larson rooted trees?

I am an undergraduate student doing a project on rooted trees. I was wondering if anyone would know any easy to understand references that explains Grossman and Larson's Hopf Algebra on rooted trees? ...
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2answers
64 views

Finding the maximum length of a minimum spanning tree

Graph G has 4 vertices/nodes and 5 edges. It is also connected. Its edges have the following weights: 5, 8, 10, 16, 18. The minimum length for a minimum spanning tree of graph G would be ...
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1answer
70 views

Complete Graph with odd degree

It is known that the Complete Graph $K_n$ has $n^{n-2}$ spanning trees. The $K_{10}$ has $10^8$ spanning Trees. Now my question: How can I compute the number of spanning Trees with odd degree of its ...
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1answer
27 views

Evaluating an arithmetic abstract syntax tree [closed]

This might be very simplistic for this forum (apologies, if so). How does one evaluate the tree below. I am not looking for a solution, just assistance in how to evaluate it: I assume I start at ...
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1answer
32 views

Find a DFS,BFS spanning tree.

Is my answer right? I think I understood the definition of BFS and DFS spanning tree, but I'm not sure my answer is right. If it is wrong, please correct it.
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2answers
43 views

Permutation of keys inserted into a tree?

Give the fraction of permutations of the keys $A $ through $G$ that will, when inserted into an initially empty tree, produce the same Binary search tree as does $A$ $E$ $F$ $G$ $B$ $D$ $C$ ANSWER: (...
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1answer
34 views

Tree properties

I am reviewing Diestel's Graph Theory and we are asked to prove that the following are equivalent: (i) $T$ is a tree. (ii) Any two vertices of $T$ are linked by a unique path in $T$. (iii) $T$ is ...
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42 views

maximizing alpha-beta puning.

I was searching a more pertinent place to post artificial intelligence concerned question, but some results pointed me to similar questions posted here, thus I chose math.se, now let's get through ...
2
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1answer
47 views

Amalgam of trees

Definition A tree is a partially ordered set $(T, <)$ such that for each $t \in T$, the set $\{s \in T : s < t\}$ is well-ordered by the relation $<$. For trees $(T,<_T)$, $(S,<_S)$, $...
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0answers
21 views

Red-black tree insert

I'm currently trying to figure out this exercise (sorry for link to image, it's for the red-black trees): http://i.imgur.com/IKMCkVf.png And I do know that the correct one is number three from the ...
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1answer
19 views

How many minimum spanning tree of following graph is possible.

How many minimum spanning tree of following graph is possible. My attempt: I've tried it manually as : Therefore, Total possible number of minimum spanning trees are $=2\times2\times2+...
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191 views

Pythagoras tree bounding size

The Pythagoras tree is a fractal generated by squares. For each square, two new smaller squares are constructed and connected by their corners to the original square. The angle of the triangle formed ...
2
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1answer
58 views

Counting the numnber of (labelled and unlabelled) rooted trees on $n$ vertices with height $h$

As far as I know, the number of labelled rooted trees on $n$ vertices is $n^{n-1}$. Is there a known result for counting the number of (labelled and unlabelled) rooted trees on $n$ vertices having ...
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0answers
34 views

Number of ordinal trees (aka rose trees) with n nodes, of depth d, with l leaves [closed]

Is computing the number of ordinal trees (also known as "Rose trees") with $n$ nodes, of depth $d$, with $l$ leaves an open problem? I assumed at first that it was a known results but I could not ...
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0answers
16 views

Terminology for property of two branches of a tree

Consider a tree $T$. A branch $B$ of a tree $T$ is just a proper subtree of $T$ (that is a subtree $B \subset T$ and $B \neq T$). Lets consider $B_1$ and $B_2$, two branches of a tree such that $B_1$ ...
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1answer
33 views

Find total number of ways to disconnect the following graph

Find total number of ways to disconnect the following graph: $4$ $5$ $6$ $8$ My attempt: I've done manually to find possible disconnected sets of given graph. I guess it is should be $8$. ...
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0answers
34 views

Assume that $|V | = |E| + 1$ and that $G$ is connected. Prove $G$ is a tree. [duplicate]

Let $G = (V, E)$ be a finite graph. (A) Assume that $|V | = |E| + 1$ and that $G$ is connected. Prove $G$ is a tree. (B) Assume that $|V | = |E| + 1$. Find an example that $G$ is not a tree.
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0answers
57 views

Enumerate out-trees that include a set of nodes in a directed graph

Given a digraph A, and an N set of nodes in the digraph. I need to enumerate the set of out-trees that contain those nodes. Where all the the out-trees leaves terminate on a node in set N. EDIT: I am ...
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0answers
14 views

Algorithm for finding all maximum out-trees in a digraph

If we have a directed graph, and the graph contains subgraphs which are out-trees. We could find the set of out-trees, such that it does not contain any out-tree that is contained by another out-tree. ...
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0answers
21 views

KD-Tree implementation with lat/lon coordinates

I have implemented a KD-Tree that stores coordinates (latitude, longitude). I have also implemented a Nearest Neighbor search algorithm using the Haversine distance. My question is, will I get correct ...
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1answer
25 views

Which of the following is NOT true for $G$?

$G$ is a graph on $n$ vertices and $2n−2$ edges. The edges of $G$ can be partitioned into two edge-disjoint spanning trees. Which of the following is NOT true for $G$? For every subset of $k$ ...
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0answers
85 views

Existence of $\lambda^+$ Aronszajn trees when $\lambda$ is regular and $2^{<\lambda}=\lambda$

While I was dealing with Aronszajn trees I found the following exercise from Kunen's old book. If $\kappa=\lambda^+$ and $\lambda$ is a regular uncountable cardinal and $2^{<\lambda}=\lambda$ ...
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1answer
17 views

Binary Heap Question interpretation

In the question in the link below, how do I tell weather to draw out a binary min heap or a binary max heap? Am I misunderstanding the question? Binary Heap Question
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1answer
27 views

Fundamental group of cylinder -triangulation method

Is this correct? Can we conclude that the fundamental group is trivial since there are no remaining generators on 1-simplices?
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1answer
20 views

Fundamental group Klein Bottle triangulation

I have been trying to find the FG of the Klein bottle, and I was wondering if someone could verify that this process is correct. After triangulating it, I then found a maximal tree (shown in yellow) ...
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1answer
39 views

Fundamental group of the sphere via triangulation

I know that the fundamental group of the sphere is zero, i.e. $\pi(S^2)=0$ I want to show this by triangulation, i.e: Triangulate the sphere Draw maximal tree Draw maximal contractable subspace ...
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1answer
41 views

Fundamental group of a tree?

Find the fundamental group of the space $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$. $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$ where $T$ is a graph $T$ is the graph made of $3$ edges with a ...
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1answer
18 views

What is the maximum number of “root subtrees” that a tree can have?

Let $T=(V,E)$ be a directed rooted tree with root $r \in V$. A root subtree$^1$ of $T$ is a directed rooted tree $T'=(V',E')$ that fulfills the following conditions: $T'$ is a subgraph of $T$, $r \...
1
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1answer
58 views

How to generate (recursively?) all non-isomorphic trees with 2 types of vertex labels with degree restrictions?

I am not sure if the title makes a whole lot of sense, but what I am trying to do is generate all non-isomorphic trees that obey the following: 1) Each vertex (including leaves) has one of two labels ...
3
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1answer
33 views

Best way to find if a subgraph has a cycle

I am implementing Kruskal's algorithm to find a minimal spanning tree of a connected graph $G$. If $H$ is a subtree of $G$, does anyone know a smart way of checking if $H+e$, where $e$ is an edge of $...
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9 views

Minimum spanning trees

Let $G = (V,E)$ be an directed connected graph with $|E|>1$, $w:E→R$ be an edge-weight function and $k∈{1,2,...,|E|-1}$. Sort the edge in E as $e_1,e_2,...,e_|E|$, such that $w(e_1) \leq w(e_2) \...
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1answer
27 views

In Graph to tree: name of operation where edges removed and vertex/edge additions?

The graph has tree paths IN-1-OUT, IN-2-OUT and IN-3&4-OUT between IN and OUT in the left. I want to make each path to a branch like the right. What is the name of this operation or the name ...
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5answers
65 views

Prove by induction: A tree on n≥2 vertices has ≥2 leaves

This is what I have. I'm pretty sure this is quite incorrect, but am I at least headed in the right direction? Base Case: $P(2)$: Tree on 2 vertices can only have one edge, the edge connecting the ...
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2answers
35 views

How to efficiently create balanced KD-Trees from a static set of points

From Wikipedia, KD-Trees: Alternative algorithms for building a balanced k-d tree presort the data prior to building the tree. They then maintain the order of the presort during tree construction ...
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0answers
26 views

Is the path from u to v is diameter of the tree?

The diameter of a tree is the longest (simple) path in the tree. Let u be a vertex in a tree and let v be the farthest from u vertex in T. Show that the path from u to v may not be a diameter I am ...
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0answers
17 views

Minimal spanning trees in multigraphs with constraints

I have a multigraph G whose edges have three identities. Let's say I have three colors of the edges red, blue and green and each two nodes may be connected by a red, blue and/or green edges. The graph ...
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1answer
32 views

Is there an efficient algorithm to find all the maximum matching in any tree?

A matching in a graph (G) is a set of mutually non-adjacent edges of (G). A maximum matching is a matching maxima cardinallity. A tree is an acyclic connected graph. Is there an efficient algorithm ...
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0answers
12 views

Minimum number of pages in a B-Tree of order n?

How can I calculate the minimum number of pages in a B tree of order n and height 3? 1st level = 1 page 2nd level = 2n+1 pages 3rd level = ? 3rd level is 1? Or how many pages?
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1answer
19 views

Completion of acyclic sub graph

Statement: Given an acyclic subgraph of a connected graph, show that this subgraph can be completed into a spanning tree of the graph. I know that there is a theorem that states that any connected ...
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2answers
58 views

Difference between Depth first search and Breadth first search algorithm

Currently I am studying Depth first search algorithm and Breadth first search algorithm. Both these algorithms are looking quite similar to me except for some differences. In BFS, we start with a ...
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0answers
14 views

Minimum Path Cover in Trees

We have a Tree with $m$ nodes and $m-1$ edges. We are given $Q$ queries. Each query consists of a list of nodes of size $k$, $[n_1, n_2, ..., n_k]$. I need to answer the minimum number of paths that ...
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0answers
23 views

Nodes lying on Same Path in Trees

Given a Treen with $n$ nodes and $n-1$ edges, I have to answer $Q$ queries. In every query, I get a list of nodes of size $k$, $n_1, n_2, ..., n_k$. I need to answer the minimum number of paths that ...
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1answer
18 views

Binary Decision Trees

I know the basics to a binary decision tree, but this problem has me a little stumped, and I'm looking for some verification on my ideas. The problem is: "Create a binary decision tree that reflects ...
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1answer
30 views

Constructing a computably infinite tree with no computable infinite branches using PA

Define an infinite tree as any set of sequences closed under prefix restriction, i.e. any prefix restriction of a sequence in the set is also in the set, where a prefix restriction is a restritcion of ...
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1answer
30 views

Show that in a binary tree, if B is the number of branch points (including the root) and L is the number of leaves, then one has the relation L = 1+B

We have been discussing trees lately, but have yet to even touch on the topic of a binary tree. I understand what a leaf is, but we didn't have one for the term "branch points" Without being 100% sure ...
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1answer
50 views

How to prove that each edge of tree is a bridge?

How to prove that each edge of tree is a bridge? My attempt: Tree is a connected graph which has no cycle, and in a connected graph, bridge is a edge whose removal disconnects the graph. Let G ...
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1answer
28 views

Question about theorem with trees

I know the theorem: for an undirected graph on $n$ nodes, any of the following two imply the third: $G$ is connected $G$ does not contain a cycle $G$ has $n-1$ edges (source) ...
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0answers
18 views

Minimum biconnected graph from spanning tree algorithm

I'm thinking about if there exist an algorithm, which can build a biconnected graph from spanning tree? The problem is that it should be minimal-weight (each edge between every pair of nodes has it ...