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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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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2answers
378 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 ...
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1answer
37 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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19 views

Let T be the resultant BFS tree. If (u,v) is an edge of G that is not in T, then which one of the following CANNOT be the value of d(u)−d(v)?

This is MCQ of a competitive exam (GATE) , answer is option (d) given by GATE . I found explanation for (d) . With the commonsense or guessing the will be option (d) generally , I have doubt "Is given ...
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3answers
886 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
104 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
49 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
25 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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16 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 ...
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2answers
4k views

Finding number of homeomorphically irreducible trees of degree N

There is a scene in Goodwill Hunting where professor challenges students with task of finding all homeomorphically irreducible trees of degree 10. This is discussed in many places, such as here and is ...
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22 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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27 views

Proof for maximum number of leaves in a tree with a given hopping distance

Hi I need help to prove the following for tree graphs which I believe is true: A tree with hopping distance $k$ (i.e. the most number of edges that any two vertices are apart) and n leaves either has ...
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2answers
191 views

Number of binary search trees on $n$ nodes of height up to $h$

How can I find the number of binary search trees up to a given height $h$, not including BSTs with height greater than $h$ for a given set of unique numbers $\{1, 2, 3, \ldots, n\}$? For example, if ...
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0answers
37 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 ...
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11 views

Completeness in M-ary trees where the value of M is variable.

Definitions of complete trees are typically limited to some specific kind of tree, often an $m$-ary tree, where the number of children each internal node must have is a positive integer $m$. Consider ...
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1answer
16 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 ...
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2answers
3k views

Show that there's a minimum spanning tree if all edges have different costs

Show that there's a unique minimum spanning tree (MST) in case the edges' weights are pairwise different $(w(e)\neq w(f) \text{ for } e\neq f)$. I thought that the proof can be done for example ...
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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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1answer
36 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 $ ...
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2answers
172 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 ...
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0answers
31 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
36 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 ...
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2answers
58 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 ...
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1answer
45 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 ...
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4answers
31 views

sum of heights in perfect binary tree

the question: what is the sum of heights of the vertices of a perfect binary tree (n vertices, the height of a leaf is 0)? explain shortly. a. $\theta(logn) $ b. $\theta(nlogn) $ c. $\theta(n) $ d. ...
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0answers
30 views

Possible Paths in Pipe Network

I'm working on this project for an oil and gas company. One of the main features is a visualization of their pipe network. I'm trying to create a tree of all possible paths. The only limit i have to ...
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1answer
42 views

how to define this directed graph satisfying these conditions?

I want to know the definition of a type of directed graph that satisfies these conditions: 1) this is a directed graph; 2) there is a directed spanning tree in this graph; 3) there is not any ...
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1answer
15 views

Vertices of RSMT

I've been looking into RSMT trees recently. For those unfamiliar with them, it's the smallest possible tree that connects a set of points using only vertical and horizontal edges. One of the ...
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0answers
20 views

Minimum spanning tree of this graph

I'm trying to find a minimum spanning tree for this graph below using Krusal's and Prim's algorithm. This is what I got for each algorithm: Krusal: visited= ...
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1answer
394 views

Binary Tree and Overhead fraction Caluculation

Find the overhead fraction (the ratio of data space over total space) for each of the following binary tree implementations on n nodes: 2) Only leaf nodes store data; internal nodes store two child ...
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14 views

Lower bound on the number of nodes in a subtree of a red black tree

Can someone give a direct proof (NOT an inductive proof) showing that a subtree rooted at any node $x$ in a red black tree has at least $2^{bh(x)}-1$ internal nodes ? $bh(x)$ means the black height ...
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1answer
34 views

how to define a “directed spanning tree”?

In all my books and articles about "graph theory", I didn't find the definition of "directed spanning tree". Could you please give this definition and the reference? How to judge if a directed graph ...
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21 views

Notation for “set of leaves of a tree” when leaves are “repeated”

Having this tree, I need to specify the number of leaves (6) and the set of leaves. Question: is this the correct notation to specify the set of leaves when there are some repeated or is there ...
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43 views

How to mathematically judge if there is a spanning tree in a graph?

Given a graph $G=(V,E,A)$ where $V$ is the set of the vertices, and $E$ is the set of sides, and $A$ is the adjacency matrix of dimension $n\times n$. $G$ is undirected or directed. We define the ...
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0answers
19 views

All possible depth first spanning trees of a directed graph.

I am looking for an algorithm that generates all possible depth first spanning trees of a directed graph that has a known root.
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1answer
25 views

How to find right child in a pyramid number?

A pyramid number: 0 1 2 3 4 5 6 7 8 9 So is there any equation like: ...
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1answer
35 views

How many ways can I connect labeled trees into a tree.

Suppose I have the labeled trees $T_{1}, \ldots, T_{n}$ with $b_{1}, \ldots, b_{n}$ vertices respectively. I would like to know how many ways I can compose a tree from these trees by using all trees? ...
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29 views

Standard notation for the set of children of a node in a rooted tree

In graph theory, given a rooted tree $T$ and a node $a \in V(T)$, is there a standard way to refer to the set of all children of $a$? I have seen $CHILDREN_T(a)$ being used, but this seem quite clumsy ...
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0answers
57 views

Minimum spanning tree for a weighted square grid

I have a particular grid with weighted edges connecting each vertex: From this I'm looking for an easy method to obtain a Minimum Spanning Tree. I can easily check columns or rows and remove all ...
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2answers
48 views

How many trees does a forest with n vertices and m edges contain?

Concerning trees in graph theory: How many trees does a forest with $n$ vertices and $m$ edges contain? This has to do with combinatorics apparently but I'm struggling with these assignments ...
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40 views

constructing directed graphs using leaves of a non-isomorphic directed binary trees

I have a simple binary tree with 4 leaves: a / \ b c / \ / \ 0 1 2 3 I want to find an algorithm that constructs all directed graphs using the tree ...
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1answer
52 views

Maximum nodes in AVL tree with distinct positive integers

Assuming that all keys in an AVL tree are distinct positive integers. Suppose that the root node of an AVL tree T holds the key N. What can be estimated largest possible number of nodes in T ? We ...
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0answers
25 views

Is there a polynomial time algorithm for Poly-trees (oriented trees) isomorphism?

In terms of graph isomorphism complexity classes Trees have a polynomial time algorithm and Directed Acyclic Graphs (DAG's) do not (so far). What about Poly-trees (oriented trees)? These are DAG's ...
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3answers
79 views

How to find the number of spanning trees for a cube?

Can you tell me a way of finding the total number of spanning trees in a $Q_d$ undirected labelled graph for $d = 3$. I know that the answer is 384, but the way (I know there are many.) of finding ...
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1answer
36 views

How to determine lexicographically the smallest Prüfer-Code of a spanning tree?

First, lexicographically the smallest means e.g. 112 < 121 and 121 < 211. EDIT: Then how to determine the minimal Prüfer-Code of a spanning tree from the given graph: Should I first find ...
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3answers
28 views

Proving the number of leaves is larger by at least two than the number of vertices with a degree of at least 3

Prove that in every tree, the number of leaves is larger by at least two than the number of vertices with a degree of at least 3. Trying induction, I get something that is too short to be right, ...
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1answer
20 views

Prove that in the union of two trees there exist a vertex with degree of at most $3$

Let $T_1=(V, E_1), T_2=(V,E_2)$ be trees on the same set of vertices, and let $G=(V,E_1 \cup E_2)$ be the graph resulting from the union of the two trees. Prove that there exist a vertex with ...
3
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0answers
53 views

Condition for a graph to have only one MST (Minimum Spanning Tree)?

Can somebody tell me if there is a condition for an edge-weighted graph to have exactly one MST? I know that it can have more minimum spanning trees, but can it have only one? Thanks in advance!
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2answers
37 views

Algorithm to cut the graph into a tree.

Given a finite connected graph $G$, I can make a finite number of cuts on the edges to obtain a tree. What is the most efficient algorithm to perform this procedure? Thanks, Vladimir
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Spare storage of a tree

I can store any undirected simple graph N vertices using $b = (N-1)N/2$ bits, by creating a mask of the edges on the upper diagonal of the adjacency matrix. For example the adjacency matrix of $K_3$ ...