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

0
votes
1answer
60 views

Counting the number of trees on $[n]$

Let $T_{n}$ be the number of trees on $[n]$. Explain the identity below in terms of $T_{n}$ and prove it. $2(n-1)n^{n-2}=\sum_{k=1}^{n-1}\binom{n}{k}k^{k-1}(n-k)^{n-k-1}.$ So far I've got that ...
0
votes
1answer
25 views

Question Concerning Family of Trees

I have the following problem where I am asked to construct a family of trees (one for each $n$) that have exactly 2 leaves. I am having difficulty with this problem mainly because I cannot find a ...
1
vote
0answers
36 views

automorphism of a rooted tree

Nowadays i'm working with tree automorphisms. I couldn't find information about rooted tree automorphism concerning the root. Does an automorphism of a rooted tree fix the root or not? Logically it ...
1
vote
2answers
72 views

In a tree, is there always a sink where every longest path ends in?

Let $T$ be an undirected tree. Can we always find a leaf vertex $s$ such that every longest path of $T$ has its other endpoint in $s$? It's easy to see that every longest path passes through the ...
1
vote
2answers
97 views

Some equivalence relation from flipping binary trees

I know almost nothing in combinatorics, so this question might be very easy, or well-known. Fix a number $n$. We will consider rooted planar binary trees with $n$ leaves. We will distinguish between ...
0
votes
0answers
16 views

Measuring values at nodes of two independent but now connected trees

I am not sure if this the right forum for this question and I hope I am providing enough details on what I want to accomplish. I have an application that has multiple trees - Tree 1 - is categories/ ...
2
votes
1answer
77 views

Given an $n$ level tree with $b$ branches at each node, how many unique paths are there from the root to the leaves?

I have a tree where, at each node, it splits into $b$ branches for a total number of $n$ levels. I enumerate the paths from the root to the leaf nodes. For example, if $n = b = 2$ then I have the ...
1
vote
1answer
38 views

Connection Trees and Partition

In our lecture we just had a short excursion into the tree-world. But the professor mentioned some connection between Ramsey and König's Infinity Lemma (If $T$ is a tree of hight $\omega$ with all ...
3
votes
2answers
179 views

König's Infinity Lemma and Aronszajn Trees

I am working through the notes of my Set Theory lecture. There my professor wrote: 'Is there an uncountable $\kappa$ such that König's Infinity Lemma holds for $\kappa$? There are models where ...
2
votes
1answer
82 views

questions about binary search tree

Show that every n-node binary search tree is not equally likely (assuming items are inserted in random order), and that balanced trees are more probable than straight-line trees. How is it prove ...
0
votes
4answers
3k views

How many edges does an undirected tree with n nodes have?

The following options: a) $n$ b) $n + 1$ c) $n - 1$ d) $n(n - 1)$ e) $(n + 1)(n - 1)$ f) $\frac{n(n - 1)}2$ g) $\lceil \log_2n \rceil$
1
vote
0answers
117 views

Generating Function for edge-rooted labelled trees

Let $T_v(z)$ be the (exponential) generating function for vertex-rooted (non-plane) trees. Im trying to construct the generating function $T_e(z)$ for edge-rooted trees from this. I know the ...
0
votes
1answer
437 views

How many vertices does a complete binary tree of height 1 have?

How many vertices does a complete binary tree of height 1 has? Height 2? Height d? Any hints on how to start to tackle these set of questions?
1
vote
0answers
99 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
148 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
183 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
80 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
76 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
614 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
136 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
49 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
56 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
193 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
91 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
570 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
57 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
187 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
45 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
79 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
537 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
93 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
136 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
245 views

How many vertices of degree 3 or more can have 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
244 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
2k 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
111 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
93 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
53 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
296 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
296 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
911 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
90 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, ...
2
votes
1answer
1k 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
398 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
50 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
177 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
106 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 ...