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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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 ...
2
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1answer
88 views

A way to codify (pre-calculatate) if a one Tree Node is a descendant of another

I have a simple, 1-directional tree representing the veins in a human body. It looks somewhat like this (red dots are nodes, blood flow is always downwards, sorry for my drawing): What I need is a ...
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2answers
2k views

Proof by induction and height of a binary tree

I need some help with a simple proof. I want to know if this proof is correct: Let's define the height of a binary tree node as: 0, if the node is a leaf 1 + the maximum height of the children ...
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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
319 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
21 views

Height of quasi-complete binary tree

Let us define a quasi-complete binary tree as a rooted binary whose nodes have all two children except at most those of the penultimate level, which can have either one or two children. I read that ...
2
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1answer
56 views

Is this random binary tree finite?

Consider the following procedure for generating a random binary tree: Starting with a full binary tree (i.e., each node has either two or no children) we iterate over the leaves and (independently) ...
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1answer
67 views

How to call a tree with a single branch?

How do you call a tree with only one branch (in other words, where every vertex has maximum one direct successor)?
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2answers
75 views

Introductions to posets on algerbaic structures (Everything I need to know about them)

I need a good and complete introduction to Tree-like orders and partial orders on algebraic structures with one operations. I accept basic texts too. I'm looking for free online texts mostly because ...
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1answer
43 views

Number of trees of a certain size

Given a branching factor $b$ and a tree height $h$, a complete tree has $\sum_{i=0}^h b^i$ nodes. Define a partial tree as a sub-tree of the complete tree, with the same root. How many such partial ...
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1answer
51 views

Counting problem (should use Cayley's formula)

How many trees above $V=\{1,2,3,4,5,6,7,8,9\}$ are there, such that $deg(4)=5$? I know I should use Cayley's formula somehow.
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1answer
2k views

How to draw all nonisomorphic trees with n vertices?

I have a textbook solution with little to no explanation (this is with n = 5): Could anyone explain how to "think" when solving this kind of a problem? (for example, drawing all non isomorphic ...
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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 ...
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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 ...
2
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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 ...
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1answer
695 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 ...
2
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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 ...
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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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1answer
37 views

Why doesn't Tutte polynomial T(1,1) equal 0?

If the formula for a Tutte polynomial is: then how does T(1,1) solve for spanning trees instead of just returning a 0?
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1answer
96 views

Preorder traversal, inorder traversal, postorder traversal

a) preorder traversal b) inorder traversal c) postorder traversal Ok, a) r,j,h,g,e,d,b,a,c,f,i,k,m,p,s,n,q,t,v,w,u b) a,b,d,c,e,g,f,h,j,i,r,s,p,m,k,n,v,t,w,q,u c) ...
2
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1answer
127 views

Proving that the height of a 2-3 tree is between $\log_3 N$ and $\lg N$

I am stuck on the problem of trying to prove the upper and lower bounds of a 2-3 tree. I think the most natural recourse is to use induction. However, my instructor told me that this was unnecessary ...
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1answer
78 views

Infinite Tree Probability Question

Suppose I have 10 dollars and I'm able to make fair 50/50 bets like flipping a coin. Now suppose each bet is for 1 dollar. What is the probability that if I keep making bets until I hit 0 dollars ...
2
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1answer
176 views

Generating function for vertices distance from the root in a planar tree

I need you help to solve this problem: Consider a planar tree with $n$ non-root vertices. Give a generating function for vertices distance $d$ from the root. Proof that the total ...
2
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2answers
63 views

Construction Types or Type Constructions?

In any (simple) type theory there are base types (i.e. the type of individuals and the type of propositions) and type builders (i.e. $\rightarrow$, which takes two types $t,t'$ and yields the type of ...
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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 ...
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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 ...
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1answer
61 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 $i$ ...
2
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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 ...
2
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1answer
135 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 ...
2
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1answer
577 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 ...
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
428 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 ...
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0answers
22 views

Red-Black tree - “Insert-Delete” [on hold]

I am looking at red-black trees. Unfortunately in my lecture notes, the operations "Insert" and "Delete" are not well explained. Could you explain to me steps that we have to do for these two ...
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0answers
37 views

Delete nodes that satisfy a property

I want to write a function that takes as argument a pointer A to the root of a binary tree that simulates a (not necessarily binary) ordered tree. We consider that each node of the tree saves apart ...
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0answers
52 views

How to check homeomorphic embedding relation programmatically?

This is a follow up to this question and Deedlit's answer. I'm looking for a precise definition of the "hem?" (tree A homeomorphically embeddable in tree B?) relation, preferably in terms of a ...
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0answers
31 views

How can I infer order from partially ordered discrete sequences?

A really interesting problem that I can't stop thinking about! Have run in to this a couple of times but yet to find a smart approach to either solve or frame this problem. This is my try at ...
2
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0answers
79 views

Can “tit for tat” strategy be defined in monadic second-order logic?

Prisoner's dilema game can be represented as a game tree, which could be infinite game with corresponding infinite game (binary) tree in common case. There is well-known tit for tat strategy, which ...
2
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1answer
58 views

Shortest path between two vertex

How we can find Shortest path between two vertex in a weighted directed acyclic graph that has positive and negative weight. in O(|V|+|E|)? thanks to all.
2
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1answer
68 views

Width and height of binary tree is $\theta(n)$?

we know this definition: Given a binary tree, Width of a tree is maximum of widths of all levels. Let us consider the below example tree. ...
2
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1answer
35 views

tree structure on classes of elements in GL_2 over a field with discrete valuation

this is my first question here, so I hope I am doing it right. :) I'm currently reading a paper about the tree of GL_2 over a discretely valued field (similarly to Serre). Here's the setting: $k$ an ...
2
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1answer
59 views

Mathematics of genealogical trees

I really searched a lot but did not find anything meeting my needs: A place where questions of genealogy, especially the structural and combinatorial analysis of genealogical "trees" of descendants ...
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0answers
37 views

What are the automorphisms of an $n$-regular tree?

Let $T$ be the connected tree in which each vertex has $n$ neighbors. (So $T$ is infinite.) What is the full automorphism group of $T$?
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1answer
55 views

Number of upper sets of size $n$ in a finite tree

Consider a finite tree $T = (V, <)$, where $y < x$ means that $y$ is the parent of $x$. We assume that $T$ has a unique root $r$ that has no parent. An upper set of $T$ is a subset $S$ of $V$ ...
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0answers
85 views

Presentation of tree decompositions (and related concepts) in terms of continuous maps?

A tree decomposition of a graph $G$ is commonly defined in terms of a tree $T$ with the following structure: Each vertex $t \in V(T)$ is associated to a set $X_t \subseteq V(G)$; The union ...
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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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711 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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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
92 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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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 ...
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3answers
70 views

Why do the children of a node $n$ in a complete binary tree have indices $2n $ and $2n+1$?

The complete binary tree is breadth-first ordered 1 to $n$ where $n$ is the number of nodes. The thing I cant seem to understand is that why are the children of node $N$ always $2N$ and $2N+1$? For ...