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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Recurrence relation induction [duplicate]

Possible Duplicate: Solving the recurrence $t(n)=(t(n-1))^2 + 1$ Show that the number of binary trees of height less than or equal to $n$ is given by the recurrence \begin{align*} ...
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1answer
2k views

Number of Trees with n Nodes

I am struggling with a question that asks the number of trees that exist with x nodes and max level z. During my research I found that the number of binary trees with x nodes can be obtained by ...
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0answers
294 views

Depth-first spanning tree?

I am going to identify tree edges and back edges in an undirected graph. The graph consists of $5$ nodes, the edges between these nodes are as shown below: Suppose starting with $v_1$, after a ...
4
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3answers
159 views

A “correct” hierarchical scoring scheme?

I have a situation where we are given a set of objects each with a numeric score stating it's importance. Let's call them Level 1 (or L1) objects. There is another set of objects that are similarly ...
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2answers
664 views

“Ballot numbers” sum up to Catalan numbers

Summing certain numbers and comparing the results with OEIS, I found that $ \sum_{k=1}^n \frac{k^2}{n} \binom{2n-k-1}{n-1} = C_{n+1} - C_{n}, $ where $C_n$ denotes the $n^{\textrm{th}}$ Catalan ...
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1answer
85 views

How can the jth level of a binary tree with n nodes has problems of size $({\frac{n}{2}})^j$?

I read from a book that the jth level (starting from j=0 or the root) of a binary tree with n nodes divides a problem into $2^j$ subproblems, each of size $\frac{n}{2^j}$. I understand where $2^j$ ...
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0answers
49 views

An MST-like problem with vertex selection

Consider a planar pointset in a rectangle, where every point has a color (an integer label). We need to select one point of every color, so as to minimize the cost of a planar MST of selected points ...
2
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1answer
126 views

Finite Rooted Binary Trees

I am new to learning about finite rooted binary trees. This lemma below is from John Meiers book: Groups, Graphs and Trees. There is no aval proof in the book. I was just wondering is I could catch a ...
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3answers
339 views

Counting $k$-ary labelled trees

The (full) binary counting tree problems gives the number of binary trees can be formed using $N$ nodes $T(n)= C_n$, where $C_i$ are the Catalan numbers. The recursion form is $T_n = ...
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5answers
11k views

The maximum number of nodes in a binary tree of depth $k$ is $2^{k}-1$, $k \geq1$.

I am confused with this statement The maximum number of nodes in a binary tree of depth $k$ is $2^k-1$, $k \geq1$. How come this is true. Lets say I have the following tree ...
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2answers
1k views

About balanced and complete binary tree

I found this and I just couldn't verify it. How come it is true? The maximum number of nodes that a balanced binary tree with depth $d$ is a complete binary tree with $2^d-1$ nodes. Let say I have ...
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1answer
538 views

Oriented trees and ordered trees

I have this confusion regarding ordered and oriented trees. I know they are both rooted and in ordered trees, the order is important. So lets say I have four nodes 1,2,3,4 then it is given that the ...
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1answer
310 views

Graph Theory - Spanning Trees

Consider a graph $G$ composed of two cycles which share an edge. $C_x$ is the cycle of length $x$ and $C_y$ is the cycle length $y$, for $x,y \ge 3$. (for example, if $x = 6$ and $y = 5$, then $C_x$ ...
2
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1answer
318 views

Evaluating 'combinatorial' sum

Help me please to calculate the following sum. I have seen such kind of formulas in the papers related to combinatorics, specifically 'trees'. I am curious how to calculate or approximate this sum: ...
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1answer
168 views

Existence of a spanning tree with certain properties

Let $\Gamma$ be a finite, connected graph (multiple edges between two vertices are allowed). Fix a vertex $u_0\in V\Gamma$. Does there exist a maximal subtree (i.e., a spanning tree) $T\subset\Gamma$ ...
3
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1answer
184 views

$\kappa$-Suslin (Aronszajn, Kurepa) subtree of the complete binary tree, $2^{\lt \kappa}$

This is from chapter 2 of Kunen, Set theory: an introduction to independence proofs. Given $\kappa$ a regular cardinal, and the existence of a $\kappa$-Suslin (Aronszajn, Kurepa) tree, show that ...
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3answers
4k views

How to show that every connected graph has a spanning tree, working from the graph “down”

I am confused about how to approach this. It says: Show that every connected graph has a spanning tree. It's possible to find a proof that starts with the graph and works "down" towards the ...
2
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1answer
281 views

Proof involving a minimum weight spanning tree.

Please help with the following homework problem: Let G be an undirected graph, $v: E\to R$ and $w: E\to R$ be two weight functions on the edges of $G$. Let $z: E\to R$ be defined as the sum of ...
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3answers
420 views

Applications of the number of spanning trees in graphs

Let $G$ be a simple graph and denote by $\tau(G)$ the number of spanning trees of $G$. There are many results related to $\tau(G)$ for certain types of graphs. For example one of the prettiest (to ...
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0answers
201 views

A tree that does not satisfy: If $v$ and $w$ are vertices in $T$, there is a unique path from $v$ to $w$?

It is a strange question on a book. Give an example of a tree $T$ that does not satisfy the following property: If $v$ and $w$ are vertices in $T$, there is a unique path from $v$ to $w$. I ...
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2answers
105 views

Narrowing a Stern-Brocot tree

Say I only wanted to enumerate the rational numbers between 0 and $a$. Is there a way to "narrow" a Stern-Brocot tree to provide this? I tried keeping my left bound at "$\frac{0}{1}$" and setting my ...
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0answers
54 views

Keeping consistency in subjective ranking

I'm doing some work on a computer program that aids in ranking items which don't have a way to objectively compare to each other. As it is now, it takes each item and pairs it up with each other ...
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1answer
66 views

Looking to generalize a binomial tree with some constraints.

I've got a set of sample data and I'm looking to see if it's possible to generalize a binomial formula to give a closed form solution to this. If not, would it be possible to write a program to do ...
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3answers
924 views

Graph with cycles proof questions

Two questions I'm stuck with: If C is a cycle, and e is an edge connecting two nonadjacent nodes of C, then we call e a chord of C. Prove that if every node of a graph G has degree at least 3, then ...
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1answer
510 views

Prove that in every tree, any two paths with maximum length have a node in common.

Prove that in every tree, any two paths with maximum length have a node in common. This is not true if we consider two maximal (i.e. non-extendable) paths. What does this even mean?
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0answers
654 views

Minimum Spanning Tree in a Complete Graph

We generate a complete euclidean graph by taking N random points from a limited (1.0 x 1.0 square) 2D space, connecting them all together (complete graph) and giving the edges weights proportional (or ...
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3answers
1k views

Sufficient conditions on degrees of vertices for existence of a tree

I am answering a question for an assignment, but I am not sure if my proof is valid, can someone look at it for me? the question: "there is a tree with $p$ vertices. If $d_1, d_2, \dots , d_p$ are ...
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1answer
345 views

Is smallest binary tree simply root node? Or does it need to have two child nodes?

Apologies for this rather simplistic question, I've just started looking at binary trees and the material I've been provided wasn't explicit about this. Presumably a parent node of a binary tree can ...
5
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1answer
533 views

Understanding various definitions of TREE($n$) in Friedman's finite form of Kruskal's tree theorem.

I was reading the Wikipedia article on Friedman's finite form of Kruskal's tree theorem, and am interested in the large numbers TREE(n). I would like to verify TREE(2)=3 myself, but find conflicting ...
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0answers
219 views

Finding the number of spanning trees of a given height

I hope I can avoid being confusing, but here goes. I have a graph $(V, E)$, connected, undirected and with no loops. I also have an assignment of integer-valued weight to each edge of the graph. ...
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0answers
101 views

A few questions about a relationship between some integer sequences and infinite recursive trees

In his book Gödel, Escher, Bach Douglas Hofstadter defines the following two integer sequences: Hofstadter G-sequence: $a(n)=n-a(a(n-1))$ Hofstadter H-sequence: $a(n)=n-a(a(a(n-1)))$ He says ...
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0answers
91 views

How matroids can help me locating trees inside a graph?

Background I am working on a project at present involving graph analysis. I basically need to mathematically model trees inside my graph. How can this be done using Matroids? What I am looking for ...
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2answers
239 views

Category of Trees as sub-category of Category of Graphs

A tree (like a binary search tree) is a direct graph with some limitations (no cycles, connected). How can I express the category of trees as "sub-category" of a graphs? There is a way? I'm not sure ...
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1answer
419 views

Number of possible Prüfer codes

I am trying to solve the following problem in my book: (Code stands for Prüfer code) Consider labelled trivalent rooted trees $T$ with $2n$ vertices, counting the root labeled $2n$. The labels are ...
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1answer
3k views

Maximum number of distinct binary tree possible with 4 nodes

what is the maximum number of distinct binary tree is possible with 4 nodes? ans is 6 but how? acc to me it should be 14
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0answers
79 views

Embedding tree metric isometrically into $\ell_\infty$

I just started (independent) learning on metric embeddings from the Fall 2003 offering of the course at CMU. I have a limited mathematical background and alas, it made me stumble at the first exercise ...
3
votes
3answers
294 views

Cheapest spanning tree

I am trying to prove the following: Let $x_1$ be any vertex of a weighted connected graph $G$ with $n$ vertices and let $T_1$ be the subgraph with the one vertex $v_1$ and no edges. After a tree ...
2
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1answer
95 views

Showing two recurrences to be identical

I am trying to prove Cayley's formula for number of labelled trees on n vertices using multinomial coefficients. The multinomial coefficient satisfies the recurrence: $\tbinom{n}{r_1,\cdots ...
3
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0answers
148 views

Number of spanning arborescences

I am trying to prove the following result from my book: Let $G$ be a directed graph with vertices $x_1,x_2,\cdots x_n$ for which a directed Eulerian circuit exists. A spanning arborescence with root ...
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2answers
858 views

Searching a binary search tree for a specific value

suppose numbers from 1 to 1000 are saved in a binary search tree and we want to find 363. Which of the following sequences cannot be the order of elements while reaching the searched value? 925, ...
0
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1answer
160 views

How many arguments are there in a Merkle tree?

I want to calculate the amount of elements in a Merkle tree given the number of leaf elements. The number of elements at a given level n is equal to number of elements at a level n+1, divided by two ...
2
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1answer
266 views

Proof about trees

Show that in any tree there exists a node such that, if we remove this node and the edges adjacent to it, we will obtain trees which have at most n/2 nodes (the removed node is not counted ...
4
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2answers
211 views

On the number of caterpillars

A caterpillar is a tree with the property that if all the leafs are removed then what remains is a path. Could you help me to prove that there are $2^{n-4}+2^{\lfloor n/2\rfloor-2}$ caterpillar on $n$ ...
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0answers
90 views

Concerning The 'Price-Collecting Steiner Tree'

I'm a Master student at the University of Leuven, Belgium. I have to make a report of a case concerning the 'Price-Collecting Steiner Tree'. We have our model and our restrictions. We are just looking ...
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3answers
93 views

Comparison trees

You have 60 coins. You know that 1 coin is either lighter or heavier than the other coins. How many comparisons are needed in a worst case scenario to discover which coin is the false one and ...
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0answers
287 views

Certain permutations of the set of all Pythagorean triples

The fact that the set of all primitive Pythagorean triples naturally has the structure of a ternary rooted tree may have first been published in 1970: http://www.jstor.org/stable/3613860 I learned ...
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1answer
405 views

Recursive Generating function for enumerating leaf labeled binary trees

Let be B(z) the exponential generating function for the number $b_n$ of different rooted unordered binary trees with exactly n leaves labeled only at their leaves (so the internal nodes are ...
4
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3answers
658 views

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 $<t>$ where $t_n$ is the ...
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1answer
329 views

Explicit bijection between ordered trees with $n+1$ vertices and binary trees with $n+1$ leaves

What is an example of a direct bijection between ordered trees with $n+1$ vertices and binary trees with $n+1$ leaves?
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2answers
108 views

A bijection between ordered trees and ballot lists

Could you help me to prove that there is a bijection between ordered trees with $n+1$ leaves and ballot lists with $n$ A's and $n$ B's? A ballot list is a sequence of A's and B's such that all ...