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.

learn more… | top users | synonyms

1
vote
1answer
7 views

Get number of vertices when number of internal vertices is known ofr a full binary tree

But I can find a counter example: * / \ * * / \ / \ * * * * Here $k = 2$, but number of vertices is 6, and number of terminal ...
0
votes
0answers
9 views

Leftish Heap and Its Right Spine

Purely Functional Data Strutures presents the following question: Chapter 3, Question 1: "Prove that the right spine of a leftist heap of size n contains, at most, floor ( log ( n + 1) ) ...
4
votes
0answers
59 views

DFS And some operations

following question on undirected graph without weights can be solved by using DFS and in O(|V|+|E|) times. check that G is ...
1
vote
1answer
38 views

Number of rooted subtrees with m edges of a p-regular tree

I have the following question: Assume I have an infinite $p$-regular tree, that is a tree where every node has degree $p$ (so also the root should have degree $p$). Then how many subtrees containing ...
-6
votes
0answers
30 views

Proving the corollary to n= mi + 1 [closed]

QUESTION: This is literally the question: Given i, then l = (m-1)i + 1, and n = mi + 1. This is a corollary to the theorem that "Given an m-ary tree with n vertices, of which i vertices are internal. ...
2
votes
1answer
63 views

Prove that the vertex degree of a minimum spanning tree is in $\mathcal{O}(1)$

I have given a set of points $S$ in $\mathbb{R}^2$. From the this points I create a mininum spanning tree MST. The euclidean distance of the points is used as the weight for the edges. The ...
-2
votes
0answers
11 views

Volume and estimate [closed]

1 the students estimate the height of a tree using a stick 10 feet high. One member of the team lies on the gound 240 feet away from the foot of the tree.He lines up the top of the tree with the top ...
6
votes
1answer
78 views

Relax function on Bellman Ford Algorithms

In a Weighted Directed Graph $G$ (with positive weights), with $n$ vertex and $m$ edges, we want to calculate the shortest path from vertex $1$ to other vertexes. we use $1$-dimensional array $D = ...
1
vote
1answer
55 views

what is an “edge disjoint spanning tree”?

if there are n = 2 vertices in a connected graph, i am supposed to have "n/2 edge disjoint spanning trees". This means i should have 1 edge disjoint spanning tree for a n = 2 graph? My best guess ...
0
votes
1answer
12 views

Graph Theory: proof about the number of vertices in a Tree's component

I'm having some problem understanding the question below: Let T = (V,E) be a tree. Show that T has a vertex v such that for all e that exists in E, the component of T-e containing v has at least ...
1
vote
0answers
32 views

How to find the number of connected components of a graph by using its 16x16 adjacency matrix?

Good day, I have this exercice that provides me with the 16x16 matrix of adjacency of a graph and it asks me to find the number of connected components of the graph and I need to give a spanning tree ...
3
votes
0answers
89 views

Associativity of the tensor product of dendroidal sets

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$. ...
3
votes
2answers
34 views

Rooted Tree and Greedy Algorithms

In a Rooted Tree, we have a message on Root. in each step, each node that has a one copy of message, can transfer this message to at most one of it's childeren. we want to use minimum step and send ...
-1
votes
0answers
13 views

Finding the wrong weight [duplicate]

I have 11 balls of the same weight (unkown) and 1 ball of different weight (heavier or lighter). If i mix these balls together - Given a Balance Scale and 3 TURNS how to find which is the odd ball ...
2
votes
1answer
94 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
votes
0answers
47 views

Graph Algorithm and Cycle Detection

In $O(|V|+|E|)$, we can detect whether a Directed Graph has a cycle or not. ---> True In depth-first seach on DAG, there is no Back Edge. ---> True With known Number of Edges, in $O(|V|)$ and not ...
1
vote
1answer
52 views

Remove edge from tree, number of vertices

Prove that if $T$ is a tree on at least $k+1$ vertices and max degree at most $d$, then there exists an edge $e$ such that the removal of $e$ causes $T$ to split into two trees where at least one of ...
2
votes
3answers
51 views

In a Tree, show that the largest degree of a node <= number of nodes of degree 1

Let $T$ be a tree in which the largest degree of a node equals to $t$. Let $n_1$ denote the number of nodes of degree $1$ in $G$. Prove that $n_1 ≥ t$ I understand why this works but I am not sure ...
1
vote
2answers
110 views

In binary tree, number of nodes with two children when number of leaves is given

For a binary tree what is the number of nodes with two children when the number of leaves is 20? I know that for complete binary tree, when the number of leaves is x then the number of internal nodes ...
2
votes
0answers
34 views

Counting unlabeled and non-uniquely labeled trees

I recently learned about Cayley's formula, which states that the number of trees on $n$ labeled vertices is $n^{n-2}$. As I understand it, this works because we can prove that there are $n^{n-2}$ ...
0
votes
1answer
33 views

Prove by induction that every complete $k$-ary tree of depth $n$ has $(k^{n+1}–1)/(k-1)$ nodes for all integers $n\ge 0$, where $k\ge 2$.

A strictly $k$-ary tree is a $k$-ary tree (a binary tree is a $2$-ary tree) in which every node has either no children (is a leaf) or $k$ children. A complete $k$-ary tree of depth $n$ is a ...
0
votes
1answer
30 views

Huffman coding - conditions for perfect tree output

The question is: Given 4 characters and their frequencies, what's the max possible difference between the frequency of the rarest character and that of the most common character, so the output Huffman ...
2
votes
0answers
70 views

Examples of forcings which add no “definable” Aronjain tree

Maybe a bit board question but: Fixing a regular cardinal $\kappa$ in the ground model, I am looking for examples of set forcing notions which preserve regularity of $\kappa$ and add no new $\kappa$ ...
6
votes
7answers
328 views

EGF of rooted minimal directed acylic graph

I am trying to find the exponential generating function of directed minimal acyclic graphs (which I now call dag), where every non-leaf node has two outgoing edges. Context: A simple tree ...
-1
votes
2answers
35 views

Binary search tree. Counting.

How many BSTrees can be constructed from given set: $\{1,2,3,4,5\}$? I have no idea how to solve it. Please help me. Thanks in advance.
1
vote
1answer
43 views

Induction Proof Check: For a binary tree T, Prove that the number of full nodes in T is always one less than the number of leaves in T.

This is a slight variant on a very common beginner's problem. I think I've got it figured out, but I wanted to make sure I actually proved what's being asked. We define a binary tree $T$: (a) A tree ...
0
votes
0answers
10 views

Changements that have to be done in order to delete node of red-black tree

According to my lecture notes: Let $x$ be the child of the node that we delete. Let $w$ be its sibling node and $p$ the father of $x$. There are four cases: At the first case, $w$ is red. We ...
3
votes
1answer
28 views

Identifying Binary Search Trees from their Prufer Sequence

If you ignore its root, a Binary Search Tree generated by some permutation of $\{1, \ldots, n\}$ is a labeled tree. Which means you can calculate its Prufer Sequence. I did this in Python and I found ...
2
votes
0answers
25 views

Red-Black tree - “Insert-Delete” [closed]

I am looking at red-black trees. Unfortunately in my lecture notes, the operations "Insert" and "Delete" are not well explained. Could you explain to me steps that we have to do for these two ...
1
vote
1answer
32 views

Determine if there is a node in a binary postorder anti-sorted tree with key $k$

A binary postorder anti-sorted tree is a binary tree for which the post-order traversal gives the keys that are saved at the nodes of the tree in descending order. Present a pseudocode for the most ...
0
votes
1answer
19 views

Is these Trees isomorphic or not?

Is these Trees isomorphic or not? They have same structure but they have different code. Because one of them is minimum code. Thank you for your answers in advance.
0
votes
1answer
31 views

Draw a 2-3 tree, insert and delete a key

Assume that at the nodes of a 2-3 tree, the following keys are saved (in an increasing order): $3,6,9,12,15,18,21,24, 27, 30, 33, 36$. It is also given that the root is a 2-node that contains the ...
1
vote
1answer
53 views

Discrete math - Prove that a tree with n nodes must have exactly n - 1 edges? [duplicate]

I'm new in discrete math. Can someone prove simply that a tree with $n$ nodes must have exactly $n - 1$ edges. I have researched the solution but I haven't founded yet. I know of course, a tree with n ...
1
vote
1answer
22 views

Rotations after inserting element in AVL-tree

We want to insert $58$ at the following AVL-tree and then we have to make rotations so that the tree is balanced. According to my notes, we are at the case RL (The first edge leads to the right and ...
0
votes
3answers
21 views

Proving if $G$ has no cycles but by adding one edge between any two vertices will create a cycle then $G$ is a tree

Prove: if $G$ has no cycles but by adding one edge between any two vertices it will create a cycle then $G$ is a tree. Below is the definition we use for a tree. I don't see any way to connect ...
2
votes
1answer
25 views

Height of quasi-complete binary tree

Let us define a quasi-complete binary tree as a rooted binary whose nodes have all two children except at most those of the penultimate level, which can have either one or two children. I read that ...
0
votes
1answer
42 views

Let $G$ be a connected graph, then $G$ is a tree iff $G$ has no cycles

Prove the following: Let $G$ be a connected graph, then $G$ is a tree $\iff$ $G$ has no cycles. $\Rightarrow$ If $G$ is connected and a tree then by the definition of tree it has no cycles. ...
2
votes
0answers
38 views

Delete nodes that satisfy a property

I want to write a function that takes as argument a pointer A to the root of a binary tree that simulates a (not necessarily binary) ordered tree. We consider that each node of the tree saves apart ...
1
vote
1answer
12 views

Prove in any tree with n vertices, the number of nodes with 3 or more neighbors is at most 2(n-1)/3

I know that the number of edges in the tree is n-1, and by the sum identity, the degree is 2(n-1)... I'm not sure how to go about completing the proof, or even starting it for that matter.
2
votes
1answer
23 views

Graph with fixed amount of spanning trees

"Find a graph with 8 vertices, which have exactly 27 spanning trees." How do I find such a graph, or prove one does not exist?
0
votes
0answers
22 views

Isomorphism testing for minimal SP-trees

I'm doing a bit of research on the SP-trees. I'm still new to this whole problematic, so I'd be thankful if someone cleared this thing up. :) This is the scenario that I'm trying to come up with a ...
0
votes
0answers
46 views

$T(n) = T(n/3) + T(2n/3) + cn$ - recursion tree with constance $c$

I have a task: Explain that by using recursion tree that solution for: $T(n)=T(\frac n3)+T(\frac 2n3)+cn$ Where c is constance, is $\Omega(n\lg n)$ My solution: 1. Recursion tree for $T(n)=T(\frac ...
0
votes
0answers
11 views

$k$-ary labeled trees with distinct labels

Classical definition of $k$-ary labeled trees doesn't restrict somehow the uniqueness of tree labels inside its branches. My question: Is any special definition (name) for such trees? To clarify ...
4
votes
0answers
22 views

Simple criteria to know if the p-nary notation of an integer can generate a tree by preorder traversing?

I am treating with a preorder tree traversal structure(which means sequences where the children of each tree node are listed behind it) now for some other problems and the structure is like: ...
0
votes
0answers
31 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 ...
1
vote
1answer
31 views

If the inorder traversal of a binary tree produces ordered output, is the tree a binary search tree?

Given a binary search tree, it's easy to see that the inorder traversal returns values from the underlying set in order (according to the comparator that set up the binary search tree). My question ...
1
vote
1answer
13 views

Proof of a tree with a vertex of degree k and less than k vertices of degree 1

The question is : Does there exist a tree with a vertex of degree k and less than k vertices of degree 1? I tried a lot but it is impossible to find. There is no tree with a vertex degree k and less ...
0
votes
1answer
37 views

Prove graph is a tree

Prove: Graph is a tree if and only if every pair of two distinct vertices are joined by only one path, and the vertices of this path are all distinct. I can easily see on a diagram that this is ...
0
votes
1answer
49 views

What does $ \chi(Tree)\leq 2 $ mean in graph theory?

I am reading an article about graphs in English. Does $\chi(Tree)\leq 2$ mean that each node has no more then $2$ children?
0
votes
0answers
32 views

Efficient and Elegant Subword-Tree Construction - trouble understanding compact representation

I'm reading the paper "Efficient and Elegant Subword-Tree Construction" by M.T. Chen and J.I. Seiferas and I'm having trouble understanding their compact representation to a Subword-tree, especially ...