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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1answer
91 views

About finiteness of trees

I am reading a book of Michael Sipser "Introduction to the theory of computation", and there is a theorem, which he gives without a proof: "If every node of a tree has only finitely many children and ...
2
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3answers
495 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 ...
3
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2answers
118 views

Bounds on how far away most leaves are from the average height of a binary tree

I'm wondering if anyone can help me prove (or disprove) this statement? Say there is a rooted full binary tree (each non-leaf node has exactly two children) with a height of $h$, an average height ...
0
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1answer
494 views

Proving by induction

I'm having a problem relating to proving by induction that the Preorder(T) and Postorder(T) algorithms both print out all the nodes in the tree without repetition. I'm not quite sure where to start.. ...
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2answers
279 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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2answers
534 views

Explanation of why the height of a binary tree $\theta({lg}(n))$.

From Heap Sort chapter of Introduction to algorithms : Since a heap of n elements is based on a complete binary tree , its height is $\theta({lg}(n))$. I know this is correct but how can this ...
0
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1answer
79 views

Number of leaves in a tree that represents a kind of permutations

Consider the following rooted tree, each of whose vertices (except for the root) is labeled with an integer $\in\{1,\dots,n\}$: let $s(v)$ be the sequence consists of the labels on the path from the ...
6
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1answer
610 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 ...
3
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0answers
66 views

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*} ...
4
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3answers
784 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 ...
4
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3answers
163 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 ...
1
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1answer
3k 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
422 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 ...
1
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1answer
68 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 ...
2
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1answer
323 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: ...
2
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1answer
141 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 ...
5
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3answers
752 views

How do you calculate the average length of a random binary tree?

Assuming that you start out with a root node, and decide with 50% probability whether or not to add two children nodes. If they do, repeat this process for them. How can you find the average length of ...
0
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1answer
93 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
56 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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2answers
2k 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 ...
0
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1answer
330 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 $C_x$ ...
1
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1answer
171 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) $T\subset\Gamma$ ...
2
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1answer
294 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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0answers
204 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
117 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
764 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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2answers
2k views

Condition on degrees for existence of a tree

Here is what I need to prove: Let $d_1,d_2,...,d_n$ be a sequence of natural numbers (>0). Show that $d_i$ is a degree sequence of some tree if and only if $\sum d_i = 2(n-1)$. I know that: 1. ...
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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 ...
0
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1answer
433 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 ...
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0answers
230 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. ...
2
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0answers
107 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
94 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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1answer
478 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
3
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3answers
327 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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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 ...
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 ...
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2answers
934 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, ...
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2answers
853 views

Partition a binary tree by removing a single edge

The question is : B-3 Bisecting trees Many divide-and-conquer algorithms that operate on graphs require that the graph be bisected into two nearly equal-sized subgraphs, which are induced by a ...
2
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1answer
270 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 ...
1
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1answer
209 views

Depth distribution of normalized decision trees?

Lets work with the following inductive definition of a decision tree: 1) $\bot$, $\top$ are decision trees. 2) If $x_i$ is a variable and $T_0$, $T_1$ are decision trees then $(\lnot x_i \land T_0) ...
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0answers
92 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
98 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 ...
2
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2answers
441 views

Number of undirected trees with labeled edges, one repeating

I need to find the number of undirected trees on $n$ vertices such that the edges (and not the vertices) are labeled and exactly one label appears twice (i.e. there are $n-2$ possible labels and they ...
0
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1answer
355 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 ...
3
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1answer
530 views

Spanning Trees of the Complete Graph Avoiding a Given Tree

EDIT: I think everyone understood, but I never explicitly stated that I am looking at labeled spanning trees. Let $T$ be a tree contained in $K_n$ (the complete graph on $n$ vertices). How can one ...
3
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4answers
2k views

Need an efficient algorithm to visit all nodes of a graph, revisiting edges and nodes is allowed

Update: This is my solution with Kruskal's Algorithm, although it doesn't take into account real "path". Brute force may be the only solution. http://www.youtube.com/watch?v=VbSwwos4R2E Hi, I ...
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0answers
95 views

Identify this combinatorial construction

I am no combinateur, but I stumbled across the following construction when studying an operad arising from information theory (actually it's a special algebra of an A$_\infty$-operad). It looked ...
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2answers
535 views

Graph - Minimum spanning tree

I have a graph with a cycle ($v_1,\ldots,v_k, v_1=v_k$). Claim: If there is a cycle with 2 edges of the same weight, and they are the heaviest edges in this cycle, then there is more than one Minimum ...