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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Multiway tree to Binary Tree

A multiway tree T can be represented as a binary tree T~ by using the firstChild and nextSibling pointers. If we think of the firstChild link as being the left link and the nextSibling link as being ...
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1answer
235 views

Graph DFS, BFS and some inference

Suppose G is a connected, undirected graph with at least 3 vertexes. we know the order or visiting the vertexes in DFS and ...
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1answer
32 views

Graphs that are almost trees

What's the name of rooted trees in which arbitrary connections between vertices of consecutive levels are allowed? (The level of a vertex is its distance to the root.) I.e.: All parents of a vertex ...
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2answers
160 views

Graph and in-Degree and Drawing

We have in and out degree of a directed graph G. if G does not includes loop (edge from one vertex to itself) and does not include multiple edge (from each vertex to another vertex at most one ...
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3answers
110 views

Acyclic graph must have a leaf

It is a theorem that every acyclic graph must have a leaf, ie. A vertex with degree 1 at most. Intuitively, it makes sense as any vertex with more degree would be connected to at least 2 vertices ...
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0answers
55 views

Ordinals as Trees

I'm trying to understand countable ordinals and their tree representation. I understand that $\omega$ is the first "non branching tree" of infinite height. I also understand that the exponent of ...
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1answer
27 views

Height of the tree : $T(n) = 4T(n/4)+2T(5n/8)+T(n/8)+\theta(1)$

Let the tree described by $T(n) = 4T(n/4)+2T(5n/8)+T(n/8)+\theta(1)$ Can someone explains why the height is $\log_{8/5}{n}$ I don't know how to proceed
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1answer
57 views

Infinite Search Tree Probibility

I have a question on Search Trees. I have a balanced, infinite, search tree. If you check a node at level $l$, the probability of finding a solution at that node is $p^l$. Questions The first ...
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1answer
247 views

Prove through structural induction that a binary tree has an odd number of nodes

A full binary tree is a binary tree where every node has either 0 or 2 children. Prove that every non-empty full binary tree has an odd number of nodes. I dont know how to get started with this ...
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0answers
74 views

Proof by Induction for Splay Tree?

I'm preparing for an exam about Trees. One of the questions that appear in Mark Allen Weiss' "Data Structures and Algorithms Analysis in C++" is: Prove by induction that if all nodes in a splay tree ...
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1answer
253 views

Graph Min Cut Problem

The idea is to give an Flow Network in which the minimum cut goes through a lot of edges. So adding one unit to each edge will change the min cut. The following figure, as a counter example, shows a ...
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1answer
66 views

$k$-connected graphs containing trees

I've encountered the following problem in the book "Graphs and Digraphs" and I'm not sure how to do it. Show that every $k$-connected graph contains any tree of order $k+1$ as a subgraph.
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1answer
122 views

Complete Binary Tree [closed]

A balanced binary tree is a full binary tree in which every leaf is either at level l or l-­1 for some positive integer l. The set of balanced binary trees is defined recursively by: Basis step: A ...
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1answer
101 views

Random Binary Search Tree, expected value of nodes with two children

In class, the professor showed that using a uniform random permutation $$ X_1,..., X_n$$ (each being i.i.d.) we can construct a Binary Search Tree by inserting the values in to the tree by their ...
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1answer
52 views

Calculating the average degree/valency of vertices

If I were to let T be a tree with n vertices, what would be the average degree/valency of the vertices in T? How would I go about calculating this?
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1answer
247 views

Proving that the height of a 2-3 tree is between $\log_3 N$ and $\lg N$

I am stuck on the problem of trying to prove the upper and lower bounds of a 2-3 tree. I think the most natural recourse is to use induction. However, my instructor told me that this was unnecessary ...
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1answer
93 views

Finding a minimum weight spanning tree? [duplicate]

Letting W be the weighted graph created by taking a complete graph K5 on five vertices 1, 2, 3, 4, 5 with the weight of each edge {x,y} given by ({x,y})=x+y, How would I find a minimum weight ...
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0answers
31 views

Proof for the number of leaves for any Binary Search Tree

A property for binary trees is that the number of leaves is the number of full nodes plus 1, in other words, $L = F + 1$ where $L$ is the number of leaves and $F$ is the number of full nodes. What ...
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1answer
66 views

How would I find a minimum weight spanning tree for W?

If I were to let $W$ be the weighted graph formed by taking a complete graph $K_5$ on five vertices 1, 2, 3, 4, 5 with the weight of each edge $\{x,y\}$ given by $(\{x,y\}) = x + y$, how would I find ...
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1answer
233 views

Question about trees, Let T be a tree with n vertices

Are my answers correct to these 3 questions? Let T be a tree with n vertices. 1) What is the average degree/valency of the vertices in T? Average Degree of of ...
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1answer
141 views

How many spanning trees does the cycle graph C2014 have?

How many spanning trees does the cycle graph $C_{2014}$ have? How do I create a bipartite graph and use it to solve this problem?
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1answer
149 views

Postorder and Preorder traversal on a Binary tree

For the tree below, list the labels of the nodes of the tree according to the pre-ordering algorithm, and then re-list them according to the post-ordering algorithm. ...
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1answer
43 views

Graph theory: tree vertices

How can I calculate the number of vertices of a tree knowing he has 33 vertices of degree 1, 25 vertices of degree 2, 15 vertices of degree 3 and all other vertices of grade 4?
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1answer
1k views

Proving every tree has at most one perfect matching

In trying to prove that every tree, T, has at most one perfect matching, I came across this idea: ...
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1answer
50 views

Spanning tree with unique paths.

Let $G$ be a connected graph and let $r∈V(G)$. Prove that $G$ has a spanning tree $T$ such that for every edge of $G$ with ends $u$ and $v$, either $u$ belongs to the unique path in $T$ with ends $v$ ...
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1answer
43 views

Expected number of subtree removal in a tree.

I was solving this problem. In a gist the problem is as follows: You are given a rooted tree. On each step you choose a node randomly and remove the subtree rooted by that node and the node ...
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1answer
84 views

Normalization of data in decision tree

After reading through a few references, I have come to know that for machine learning in general, it is necessary to normalize features so that no features are arbitrarily large ($centering$) and all ...
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1answer
61 views

The intersection of $k$ subtrees of a tree $T$ is nonempty.

Let $T_1$, $T_2$, . . . , $T_k$ be subtrees of a tree such that any two of them have a vertex in common. Prove that they all have a vertex in common. Any hints/solutions are greatly appreciated. I am ...
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1answer
35 views

Find the node that we will reach with a given path on a graph (complete binary tree)

This question is regarding a special case of graph i.e. complete binary tree Consider the following tree :- ...
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2answers
487 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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0answers
28 views

AVL, keys where rotations are not required

Suppose the keys {1,2,3.....,n} are inserted into n empty AVL tree in sequence 1,2,3.....,n. Find the key values(1,2...n these are the keys) where rotation(rotations to balance the tree structure) is ...
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1answer
68 views

Preorder Traversal

For Each Preorder Traversal, we have multiple Inorder Traversal. this is True or False Conclusion? every one would help me and add some detail.
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1answer
180 views

Visiting Node in BFS and DFS in the same order [closed]

if G be a connected, undirected graph and has at least 3 vertex. we know the order of visiting node from a given vertex in BFS and DFS is the same. which of the following is false? a) G can be a ...
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1answer
69 views

Shortest path between two vertex

How we can find Shortest path between two vertex in a weighted directed acyclic graph that has positive and negative weight. in O(|V|+|E|)? thanks to all.
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1answer
46 views

A graph with minimum degree $k+2$ contains any $(k+3)$-vertex tree as a subgraph?

Let $k$ be a positive integer and let $T$ be a tree of order $k+3$. If $G$ is a graph with minimum degree at least $k+2$, prove that $G$ contains a subgraph isomorphic to $T$. Any solutions or hints ...
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0answers
34 views

Pascal's Identity and Trees

Pascal's Identity states that $n \choose k$ = $n-1 \choose k-1$ + $n-1 \choose k$ since if one element is separated from the rest we can claim that either it is chosen (resulting in $k-1$ elements ...
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1answer
50 views

Automorphism of Tree

Let $\sigma$ and $\theta$ be two automorphisms of tree $X$. I want to show that min$_{v\in V(X)}d(v,\sigma(v))=$min$_{v\in V(X)}d(\theta^{-1}\sigma\theta(v),v)$. I know every automorphism of tree is ...
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1answer
71 views

Iterations of Pascal's Identity

Pascal's Identity states that $n \choose k$ = $n-1 \choose k-1$ + $n-1 \choose k$ since if one element is separated from the rest we can claim that either it is chosen (resulting in $k-1$ elements ...
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1answer
86 views

How to understand the perfect binary tree formula?

I got this paragraph by reading "python algorithm", in which it mentioned `some knights participate in an knockout match, how many mathes do they need to produce the winner. It's answer says: I'm ...
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0answers
53 views

What are the algebraic and topological properties of a tree (graph theory), and what are any possible connections?

I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup ...
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1answer
108 views

Infinite Tree Probability Question

Suppose I have 10 dollars and I'm able to make fair 50/50 bets like flipping a coin. Now suppose each bet is for 1 dollar. What is the probability that if I keep making bets until I hit 0 dollars ...
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1answer
138 views

The union of two connected graphs is connected [closed]

Let $G = (V,E)$ be a graph and let $H_1 = (V_1,E_1)$ and $H_2 = (V_2,E_2)$ be two connected subgraphs of $G$ that have at least one node in common. Prove that the graph $H = H_1\cup H_2 = (V_1\cup ...
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3answers
99 views

How to write a summation function that counts the number of nodes in a tree?

I come from a programming background and am interested in learning how to represent some things as simple equations, as an entry into thinking mathematically. How do you represent a tree structure as ...
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1answer
126 views

Width and height of binary tree is $\theta(n)$?

we know this definition: Given a binary tree, Width of a tree is maximum of widths of all levels. Let us consider the below example tree. ...
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0answers
43 views

Tree of arity n: How to call a vertex that has only k (k<n) children?

What is the correct adjective for a vertex in an n-ary tree that has only k children (k < n)? I was thinking of something like "unsaturated", but I don't know if that is the correct word for this. ...
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1answer
72 views

Number of distinct Binary tree formed with respect to height h

How many types of distinct Binary Tree can be formed with a height of h? assuming height starts from 0 when the tree has only the root. example: if the height of tree is 1 then root-leftchild ...
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1answer
26 views

Proof about spanning tress in graphs

Let $G=(V,E)$ be a graph and $T_i=(V,F_i),i=1,2$ two disjoint spanning trees in $G$. Let $f_1 \in F_1$. Prove that there is $f_2\in F_2 $ such that $T:=T_1-f_1+f_2$ is a spanning tree.
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28 views

How does inserting $N$ objects one at a time into an ordered AVL tree yield an efficient sorting algorithim

If we assume reblalancing an AVL tree of height n after an insertion or deletion takes $O(n)$ operations. How does inserting $N$ objects one at a time into an ordered AVL tree yield an efficient ...
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2answers
127 views

Prove that if G is a tree in which all vertices have odd degree then G has odd size.

Prove that if G is a tree in which all vertices have odd degree then G has odd size. Good night, do not know how to approach this "prove". Can you give me tips to solve it?. Please.
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1answer
149 views

Formula for number of “root” nodes in a tree where Parent shares child nodes?

If I have a tree like this: {a},{b,c},{d,e,f},{g,h,i,j} in this case we have a total of 10 nodes. Is there any equation where given "10" I can calculate how many bottom nodes there are (answer: "4" ...