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

3
votes
1answer
62 views

Number of nodes with even offspring

I've been working on a combinatorics assignment, and while the last few questions had clever solutions which didn't involve functional equations and the use LIFT, I fear I'm at my end. Given a ...
1
vote
1answer
34 views

Does the Prim algorith always create the same tree despite the starting node?

Does the Prim algorith always create the same tree despite the starting node? PD: sorry for my english.
0
votes
1answer
88 views

Inserting values left to right in a binary search tree

What does it mean to build a binary search tree by inserting values from left to right starting from an empty tree? The "left to right" part confuses me..I know how to build one by normally inserting ...
0
votes
2answers
2k views

How to find non-isomorphic trees?

"Draw all non-isomorphic trees with 5 vertices." I have searched the web and found many examples of the non-isomorphic trees with 5 vertices, but I can't figure out how they have come to their ...
0
votes
2answers
60 views

Finding an equation for a growth formula

Given a tree that has three nodes each level I want to find the formula that predicts the number of all nodes with a given tree height. I fitted the data into Numbers with an exponential function ...
3
votes
2answers
552 views

Number of binary trees with N nodes

I am trying to calculate the number of trees (non isomorphic) with n nodes (total including leaves). I think that there are n! such trees, but I don't know how to prove that. I know that the number ...
2
votes
1answer
141 views

spanning trees of an edge transitive graph

Let $G$ be an edge transitive graph. Let $t(G)$ be the number f spanning trees on $G$. Show that each edge lies in exactly $\tfrac{(n-1)t(G)}{m}$ spanning trees. Where $|V(G)|=n$ and $|E(G)=m$. ...
2
votes
2answers
2k views

Can Prims and Kruskals algorithm yield different min spanning tree?

In this problem I am trying to find the min weight using the Prims and Kruskals and list the edges in the order they are chosen. For Prims I am getting order ...
0
votes
2answers
697 views

creating a Binary tree based on a prefix expression

I want to find the value of a the prefix expression -/+8,10,2*3,2 and build its binary tree I am trying to learn this for a math course, but have absolutely no clue ...
0
votes
0answers
488 views

D ary tree node math

A d-ary tree is a rooted tree in which each node has at most d children (c) Suppose the tree has n nodes. What is the minimum the depth could possibly be, in terms of n and d? You can leave your ...
2
votes
4answers
1k views

How many labeled trees exist on n vertices with exactly 3 vertices of degree 1?

My combinatorics class is covering spanning trees right now and one of the questions being asked is "What is the number of labeled trees on n vertices with exactly $3$ vertices of degree $1$?" I've ...
0
votes
1answer
63 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
39 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
79 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 ...
2
votes
2answers
105 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
92 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
39 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
210 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
92 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 ...
3
votes
4answers
4k 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
140 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
561 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
119 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
173 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
288 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
87 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
692 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
180 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
52 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
60 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
110 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
951 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
69 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
228 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
53 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
91 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
724 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
99 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
148 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
328 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
261 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
122 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
117 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
60 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 $i$ ...