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
0answers
117 views

Binary Tree and Geometric Distribution

I have the following algorithm for "constructing" a binary tree: A probability $p_g$ for elongation, i.e. adding an edge A probability $p_b$ for branching, i.e. adding to a node two "child" edges ...
1
vote
2answers
171 views

Draw Graph from distance to other nodes

I have a matrix that shows the distance from a node to another node: A B C D E A 0 2 4 3 1 B 2 0 2 1 3 C 4 2 0 2 1 D 3 1 2 0 2 E 1 3 1 2 0 To clearify: The 2 ...
1
vote
0answers
281 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 ...
0
votes
1answer
86 views

Unique solution for a given pre- and post-order of a rooted tree

Decide the picture of a rooted tree with pre-order $a,b,c,d,e,f,g,h$ and post-order $d,e,f,g,h,c,b,a$. Show that there always is a unique solution for a given pre- and post-order of a rooted tree. My ...
3
votes
2answers
77 views

Number of undirected trees

Given n numbered vertices I want to know the number of different trees that can be created with them. I know that cayley's theorem says it's $n^{n-2}$, but why can't it also be: ...
2
votes
1answer
679 views

Number of distinct path in a graph with $n$ vertices

Let $T = (V , E)$ be a tree with $|V | = n\geqslant 2$. How many distinct paths are there (as sub graphs) in $T$? I already have the answer to this question as $(n/2)$. The problem that I'm having is ...
6
votes
2answers
177 views

What is the average weight of a minimal spanning tree of $n$ randomly selected points in the unit cube?

Suppose we pick $n$ random points in the unit cube in $\mathbb{R}_3$, $p_1=\left(x_1,y_1,z_1\right),$ $p_2=\left(x_2,y_2,z_2\right),$ etc. (So, $x_i,y_i,z_i$ are $3n$ uniformly distributed random ...
2
votes
1answer
50 views

Is there a tree $T$ such that $\text{diam}(T) \geq k$, where $k$ is the number of vertices with degree less than 3?

Let $T$ be an undirected tree, let $d$ be the diameter of $T$, and let $s$ be the number of vertices in $T$ with degree less than 3. Recall the diameter of a graph is the length of the longest ...
0
votes
1answer
51 views

Getting a values from nodes

The goal: get horizontal values of vertical level N where level 1 is pinacle node (1). Example: level 4 as input should produce: | 1 | 3 | 3 | 1 | Note: the sum ...
0
votes
1answer
59 views

Correspondence between fractal sets and trees

In Hillel Furstenberg's series lectures on ergodic theory in fractal geometry, he mentioned his search on finding a one-to-one correspondence between fractal sets and trees, however, I couldn't not ...
6
votes
4answers
202 views

Recursive Sequence Tree Problem (Original Research in the Field of Comp. Sci)

This question appears also in http://cstheory.stackexchange.com/questions/17953/recursive-sequence-tree-problem-original-research-in-the-field-of-comp-sci. I was told that cross-posting in this ...
0
votes
1answer
106 views

Prove that there are two end points in a tree with a common neighbor

Let $T=(V,E)$ be a tree with at least $3$ vertices. Assume that every vertex has either degree $1$ or a degree of at least $3$ (so there are no vertices with degree $2$). Prove that there are two end ...
1
vote
2answers
878 views

Gallery of unlabelled trees with n vertices

Can anyone point me to a gallery (printed or online) of unlabelled trees, sorted according to their order (i.e., number of vertices)? That is, for each order n in oeis.org/A000055 (up to maybe n=11 ...
0
votes
1answer
66 views

Minimal Red-Black tree with depth 3

I'd like to ask what is minimal RBT with black depth 3. Is this following RBT ? Values are not important. And that tree can't have depth 2 or 1.
3
votes
2answers
222 views

In any tree, what is the maximum distance between a vertex of high degree and a vertex of low degree?

In any undirected tree $T$, what is the maximum distance from any vertex $v$ with $\text{deg}(v) \geq 3$ to the closest (in a shortest path sense) vertex $y$ with $\text{deg}(y) \leq 2$? That is, $y$ ...
2
votes
2answers
52 views

Embedding of Tree

Q. Proof for every Tree can be embedded into the plane. Conditions. We cannot use Euler Formula for Planar Graphs. We can use definition of tree, $V-E=1$, no-cycles, every edge is critical, there ...
1
vote
3answers
89 views

Parent and childs of a full d-node tree

i have a full d-node tree (by that mean a tree that each node has exactly d nodes as kids). My question is, if i get a random k node of this tree, in which position do i get his kids and his parent? ...
5
votes
2answers
710 views

How many vertices of degree 1 in a tree?

How many vertices of degree 1 are there in a tree with no vertices of degree more than 4? The only thing that I have right now is that the number of edges in a tree is n-1 where n is the number of ...
0
votes
2answers
97 views

Prove there is a tree with $n$ vertices having degrees $d_1, d_2…d_n$

For $n ≥ 2$ suppose $d_1, d_2,....d_n$ are positive integers with sum $2n - 2$. Prove there is a tree with n vertices having degrees $d_1, d_2....d_n$. I'm at a loss on this one. I'm sure it's pretty ...
1
vote
2answers
145 views

Trees with vertex set

I am having hard time understanding and solving the following question: There are exactly three trees with vertex set {1,2,3}. Note that all these trees are paths; the only difference is which ...
4
votes
1answer
322 views

How many vertices of degree 3 or more can a tree have at most?

It is known that a tree $T=(V,E)$ has at least $\Delta$ leaves, where $\Delta$ is the maximum degree of $T$. But how many vertices of specific degree at least $k$ can a tree have at most? I'm ...
1
vote
1answer
260 views

No of labeled trees with n nodes such that certain pairs of labels are not adjacent.

What is the number of trees possible with $n$ nodes where the $i$th and $(i+1)$th node are not adjacent to each other for $i \in \left[0,n-1\right)$ and $$i/2 = (i+1)/2.$$ (integer division) (nodes ...
3
votes
2answers
3k 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 ...
0
votes
2answers
120 views

Question: Graph Theory and Trees

In a group of 2n schoolchildren each one has at least n friends. On an outing, the teacher tells them to hold hands in pairs. Show that this can be done with each child holding a friend’s hand, and ...
0
votes
2answers
113 views

how to store a math problem in a binary tree?

If I have the following problem: $\ 12 - (2 +3) - (3 *4)/ (5 -7) $ How would it be stored in a binary tree? following the order of operations, would you start with $\ (3*4) $ at the top or $\ 12 $ ...
2
votes
1answer
59 views

Proof of the Converse of Kraft's Theorem

So I have already proven Kraft's theorem for ternary trees, and I have been tasked with proving the converse. That is, I need to show that there is a ternary tree with $k$ leaves, such that leaf $i$ ...
0
votes
0answers
21 views

Simplification of a dervived binary tree with n nodes [duplicate]

hi I need help with this problem how do simplify this equation and what are the steps and approaches to this problem
14
votes
2answers
333 views

Is there a “tree-like” proof of compactness theorem in the case of uncountably many variables?

I like proofs using trees and König's lemma, since they are very visual. One of the applications of König's lemma you can show to students is proving compactness theorem for propositional calculus, ...
0
votes
0answers
344 views

What is the algorithm to sort 5 elements in 7 binary comparisons?

I'm tasked with finding the algo that sorts 5 elements in 7 binary comparisons. (The 7 is derived from ceilingFunction(log 5!), which our text states is the minimum number of comparisons required for ...
2
votes
2answers
1k views

Tree pruning question…

all. I'm facing the question: "A chain letter starts when a person sends a letter to five others. Each person who receives the letter either sends it to five other people who have never received it ...
3
votes
0answers
99 views

Mathematical notation for formulas involving trees

I am working on document that requires me to write such things as "$T_1$ is a descendant of $T_0$", or "$N_1$ is an parent of $N_2$". For now, I've been highjacking set notation for use in formulas, ...
3
votes
1answer
2k views

Show that Minimum Spanning Tree is unique

Show that MST is unique in case the edge weights are pairwise different $(w(e)\neq w(f) \text{ for } e\neq f)$. I thought that the proof can be done for example by contradiction, saying that we ...
1
vote
1answer
538 views

How can I tell how many non-isomorphic unrooted trees with 6 edges exists without drawing them all?

Typically my professor asks that we draw them all, but I would like to save some time to confirm how many I need.
1
vote
0answers
52 views

Red Black Binary Search Trees

Give an example of a Red-Black tree and a value, for which inserting the value, and then immediately deleting it yields a tree that is different from the tree before the insertion.
0
votes
1answer
42 views

Identify Type of Recursive Sequence?

I would love to learn techniques for solving the following, but I can't seem to identify this type of sequence: let $N > 0$ and let $k$ be an arbitrary positive integer between $0$ and $N-1$ ...
0
votes
1answer
204 views

Let T be a tree with sub-trees which each set has a vertex in common - hence T has a vertex in all of its sub-trees?

The question is: Let T be a tree with sub-trees $T_1,T_2,..,T_n$ such that all trees $T_i,T_j$ have a vertex in common which each set has a vertex in common - show that T has a vertex in all $T_i$. ...
0
votes
0answers
109 views

Nilpotency of the adjacency matrix of a directed tree network

Say I have a directed network that is organized in a tree, with all connections going downstream (genealogically). By that I mean that there is one root node connected to $c_{00}$ child nodes, and ...
1
vote
1answer
184 views

Proofs involving subtrees of a tree

I have found some claims about trees in my graph theory text, and I am wondering if corresponding proofs can be found, as I cannot find any online or in another text. First, If $T_1$ and $T_2$ ...
0
votes
1answer
147 views

Recurrence relation for the number of spanning trees in a connected graph proof

The number of spanning trees in a graph $G$ containing an edge $e$ is equal to the number of spanning trees in $G/e$. The number of spanning trees in a graph $G$ not containing an edge $e$ is equal to ...
1
vote
1answer
257 views

What is the length of the Minimum Spanning Tree

What is the length of the Minimum Spanning Tree for the following weighted graph? Solution. The length of any minimum spanning tree for this graph (and there is more than one) is 60. The graph and ...
7
votes
1answer
1k views

Height of a full binary tree

A full binary tree seems to be a binary tree in which every node is either a leaf or has 2 children. I have been trying to prove that its height is O(logn) unsuccessfully. Here is my work so far: I ...
3
votes
1answer
109 views

Prong Corollary, $G$ has a subgraph isomorphic to $T$

There is a corollary in Diestel textbook Graph Theory. Corollary 1.5.4. if $T$ is a tree and $G$ is any graph with $\delta(G) \geq |T|-1$, then $T \subseteq G$, i.e. $G$ has a subgraph isomorphic ...
0
votes
2answers
770 views

Proving a simple connected graph is a tree if adding an edge between two existing vertices of T creates exactly one cycle

When proving a simple connected graph is a tree if adding an edge between two existing vertices of T creates exactly one cycle, is it sufficient to just remove that edge that created a cycle, then it ...
3
votes
0answers
289 views

Algorithm for generating homeomorphically irreducible trees of size n

In this video they talk about generating all the homeomorphically irreducible trees of size 10. I was wondering if there is a generating algorithm for generating all the homeomorphically irreducible ...
8
votes
2answers
216 views

Ordinal interpretation of Friedman's $n$?

I heard that Kruskal's tree theorem can be turned into a finite form that creates an extremely fast growing function because ordinals could be encoded into trees. On this wiki page it mentions that ...
1
vote
2answers
100 views

spanning trees of graphs

Assume we have a simple connected graph G, how would start a prove of the following statement? For any edge of G, there is a spanning tree of G that contans it. I have decided that this is a true ...
3
votes
1answer
60 views

What's the rank of this well founded relation?

Definition A tree is an ordered list of trees. (N.B these are finite objects and there is a very simple computable bijection of them with $\mathbb N$) Examples [] and [[],[],[]] and ...
0
votes
0answers
100 views

Straight skeleton is a tree

Can anybody give me a hint on how to prove that the straight skeleton of every polygon is a tree. Here is the definition of the straight skeleton (taken from Wikipedia): The straight skeleton of a ...
2
votes
2answers
4k views

Determining Ambiguity in Context Free Grammars

What are some common ways to determine if a grammar is ambiguous or not? What are some common attributes that ambiguous grammars have? For example, consider the following Grammar G: $S \rightarrow ...
1
vote
1answer
203 views

For BSP generation, How to intersect or locate a triangle with a plane defined by another triangle?

I've hit a stumbling block in my project to draw the Utah teapot. I want to generate a binary space partition tree of a set of 3D triangles. The decision step in the recursive tree-construction ...