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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How to find the maximum number of vertices in a tree with respect to maximum path length and maximum degree value

Given a tree, find the maximum number of vertices $v$ in that tree using the maximum path length $p$ and a maximum degree that applies to all vertices $d$. Assuming that I drew my test tree ...
6
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2answers
140 views

Question about trees and generalizing the Principle of Dependent Choices.

One form of the Principle of Dependent Choices is that for any tree $T$ of height $\omega$ such that every node of $T$ has a successor, there is a branch of $T$ of length $\omega$. In this post, I ...
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2answers
309 views

Bijection between binary trees and plane trees?

I would like to describe a bijection between binary trees and plane trees. A binary tree has a root node and each node of the tree has at most 2 children (left and right). A plane tree has a root node ...
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1answer
163 views

Depth first search on graph

I have a homework problem I think I know the answer to, but want to double check Consider the graph with three nodes, $a$, $b$, and $c$, and the two arcs $a \rightarrow b$ and $b \rightarrow c$. ...
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0answers
49 views

What is the runing time of this algorithm involving length and depth?

I'm hoping that someone can shed some light on this running time. I have a "tree", for lack of a better description, that has a length $l$ and depth $d$. I want to maximize the tree size, which ...
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1answer
208 views

has deleting node in a binary search tree Displacement feature?

I am developing an academic project about graph and tree theory.I searched a lot but I didn't find a clear answer. In a part of project we want to delete some nodes from tree for example we want to ...
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1answer
293 views

Length of Shortest Path in a Generated Binary Tree

Let's say I have a binary tree of $2$-tuples of positive integers starting with $(1,1)$. The left-child of any element $(A,B)$ is $(A,A+B)$, and the right-child of any element is $(A+B,A)$. Hence, the ...
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3answers
437 views

$n$-ary trees with $k$-internal nodes - Catalan numbers

It is known that the Catalan numbers count the number of binary trees with $k$-internal nodes. I was thinking of how to count ternary trees or in general $n$-ary trees with $k$ internal nodes and got ...
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1answer
105 views

How to establish bijective between the set of rooted trees and natural numbers, using Godel numbering?

Consider the structure of a rooted tree independent of its underlying set, (i.e. in the sense of trees as combinatorial species). I know a number of ways which we can encode any such tree in natural ...
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1answer
184 views

What is the fairest solution/formula for rewarding points in a hierarchical network?

Introduction The nature of this hierarchical network is based on the concept of Multi-Level Marketing strategy. Example 1 - Unfair Situation Ancestor receives 1 point for every descendant ...
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2answers
159 views

existence of a spanning tree

Let $T$ and $T'$ be two spanning trees of a connected graph $G$. Suppose that an edge $e$ is in $T$ but not in $T'$. Show that there is an edge $e'$ in $T'$, but not in $T$, such that ...
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1answer
801 views

M-ary tree problem

A full $m$-ary tree $T$ has 81 leaves and height 4 1) Give the upper and lower bounds for $m$ 2) What is $m$ if T is also balanced? [with $m^h=l$ for maximum leaf in a m-ary tree $m^4=81$ then m=3 ...
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2answers
227 views

Tree problem about preorder notation

Show that an ordered rooted tree is uniquely determined when a list of vertices generated by a preorder traversal of the tree and the number of children of each vertex are specified.
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0answers
70 views

Groups acting on (regular) trees with finite quotient

Let $T$ be a regular tree, and suppose that $G \leq \mathrm{Aut}(T)$ has finite quotient graph, $T / G$. Is it true (in general) that $G$ will have trivial centralizer in the full automorphism group? ...
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0answers
78 views

How can I prove this property of a $d$-ary tree?

I have the following homework (algorithms lecture): Every $d$-ary tree $G=(V,E)$ contains a vertex $v$ such that the size of the subtree with root $v$ is at least $\frac{1}{d+1} \vert V \vert$ and at ...
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1answer
176 views

Cubic (3-regular) graph spanning tree

Considering loop free cubic graphs (graphs where every node has 3 neighboring nodes): Is is possible to construct a spanning tree that only has nodes with 3 neighbors in the spanning tree or 1 ...
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1answer
425 views

Help in understanding search of Vantage-Point tree

This is my reference: http://stevehanov.ca/blog/index.php?id=130 A vantage-point tree is a way of organizing a set of points so that finding the n-nearest neighbors is as efficient as possible. It ...
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1answer
36 views

If $G$ is a tree and $\forall v_1, v_2 : dist(v_1, v_2)$ is even, where $v_1, v_2-$ are leaves $\Rightarrow \exists!$ a maximal independent set.

If $G$ is a tree and $\forall v_1, v_2 : dist(v_1, v_2)$ is even, where $v_1, v_2-$ are leaves of the tree $\Rightarrow \exists!$ a maximal independent set. Give some clue please! Thanks anyway!
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3answers
3k views

Minimum Number of Nodes for Full Binary Tree with Level $\lambda$

If the level ($\lambda$) of a full binary tree at zero is just a root node, than I know that I can get the maximum possible number of nodes (N) for a full binary tree using the following: N = ...
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1answer
81 views

Constructing a tree from disjoint graphs

I will preface my question with the definition of a simple tree that applies to my question: -"A simple tree is an undirected and connected graph with no cycles."- I am having difficulty coming up ...
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1answer
191 views

Prove equivalence of conditions for a tree

Let $G=(V,E)$ denote a nonempty graph. Show that the following conditions are all equivalent. $G$ is a tree. Any two vertices in $G$ can be connected by a unique simple path. $G$ is ...
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3answers
1k views

What is the main difference between a free tree and a rooted tree?

In graph theory what is the difference between a rooted tree and a free tree ? What is normally meant when just the plain "tree" is used ?
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1answer
84 views

Natural order of rational trees?

What would be a natural order of rational trees? Rational trees arise naturally from free algebras if we view a term as a finite tree. For example the term f(a,g(b,c)) could be viewed as the ...
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1answer
574 views

Every automorphism of a tree with an odd number of vertices has a fixed point

If $T$ is a tree, and $T$ has an odd number of vertices, then $\forall f$, where $f$ is automorphism $\Rightarrow \exists$ fixed point (vertex). What it means: Formally, an automorphism of a tree $T$ ...
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1answer
585 views

How to make a parse tree for the following propositional logic formula?

I have a formula $\neg (( q \rightarrow \neg q) \vee p \vee ( \neg q \rightarrow ( r \wedge p)))$. As it contains 3 subformulas between the $\vee$'s, how can I put it into a parse tree. Would it be ...
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2answers
508 views

Proof for Full Binary Tree Using Handshaking Lemma?

I asked a question a few days ago and figured out the proof for this theorem using induction. ...
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1answer
523 views

Proving terminal vertices and total vertices of a full binary tree?

I am trying to make a proof by induction of the following theorem. ...
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1answer
211 views

A binary tree in 3-ary tree

We have an infinite $3$-ary tree, with root $R$. In coloring $C(p)$ each edge is black with probability $p$ and white with probability $1 - p$, and edges are independent. Show that there is a ...
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2answers
205 views

Why for number of leaves in a tree (all types of trees) is it true

I have to prove the following claim, given the tree $T=(V,E)$, $|V|\geq3$: $$|V_1| \leq \frac { |V| \times (\Delta (V) - 2) + 2 }{ \Delta (V) - 1 } $$ where $|V_1| - $ number of leaves in a tree, and ...
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0answers
48 views

How to formulate a best-search algorithm limited by a count of nodes visited?

The problem I'm doing a search by computer program. Each node takes about 5 minutes of wall time to get a result so I'm looking to carefully choose the nodes to inspect so as to find the best result ...
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2answers
609 views

graph theory and forests

We were given an this question in my class: Prove that a forest with n vertices and m components has n-m edges using induction on m. Induction is not my strongest point and I was wondering if anyone ...
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1answer
165 views

Probability of passing through 3 specific nodes along a binomial tree

Consider a re-combining binomial tree with probability of up = $p$ and probability of down = $(1-p)$. Let $n$ be the number of time steps in the binomial tree (the $x$-axis is time, and each column of ...
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1answer
89 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 ...
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2answers
116 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 ...
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1answer
376 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
475 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 ...
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1answer
245 views

characteristic polynomial of the adjacency matrix of a tree

I have read that if $A$ is the adjacency matrix of a tree $T$, then we have that $$\det(\lambda I - A) = \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k N_k(T) \lambda^{n-2k} $$ where $N_k(T)$ is the number ...
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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 ...
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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*} ...
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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
348 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 ...
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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 ...
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2answers
747 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
87 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
51 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 ...
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1answer
138 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
378 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
14k 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
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 ...
2
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1answer
578 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 ...