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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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 ...
2
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1answer
92 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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1answer
274 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 ...
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3answers
207 views

What is the maximal path of a tree?

Could anyone explain obviously what the maximal path is ? Is it necessary for a tree that has two maximal paths that share no common vertex ?
2
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3answers
120 views

In a Tree, show that the largest degree of a node <= number of nodes of degree 1

Let $T$ be a tree in which the largest degree of a node equals to $t$. Let $n_1$ denote the number of nodes of degree $1$ in $G$. Prove that $n_1 ≥ t$ I understand why this works but I am not sure ...
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2answers
1k views

Tree pruning question…

all. I'm facing the question: "A chain letter starts when a person sends a letter to five others. Each person who receives the letter either sends it to five other people who have never received it ...
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3answers
699 views

Graphs: trees, induction proof

I was wondering if you could help me prove the following. $G$ is a tree $\iff$ deleting any edge will disconnect it. And a similar one: $G$ is a tree $\iff$ adding any edge will create a cycle. I ...
2
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1answer
301 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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2answers
48 views

could a spanning tree graph be expressed by a lower triangular matrix?

Suppose a directed spanning tree graph $G$, there are $n$ nodes, and the root is node $1$. We express this graph by a matrix $M_{n\times n}$. If there is an directed edge from node $i$ to node $j$, ...
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2answers
115 views

A tree on n vertices where every vertex has degree 1 or 4. Prove that n ≡ 2 (mod 3)

Let T be a tree tree on n vertices where every vertex has degree 1 or 4. Prove that n ≡ 2 (mod 3)
2
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1answer
125 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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3answers
603 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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1answer
562 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 ...
2
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2answers
658 views

Number of inner nodes in relation to the leaf number N

I am aware that if there is a bifurcating tree with N leaves, then there are (N-1) internal nodes (branching points) with a single root node. How is this relationship proved? Best,
2
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1answer
260 views

Fast question about minimum spanning trees

If any edge from a given spanning tree T0 is contained in some minimum spanning tree T*, does this imply that T0 is also a minimum spanning tree ? Right now, I'm trying to draw on paper some graphs ...
2
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1answer
93 views

Prove that the vertex degree of a minimum spanning tree is in $\mathcal{O}(1)$

I have given a set of points $S$ in $\mathbb{R}^2$. From the this points I create a mininum spanning tree MST. The euclidean distance of the points is used as the weight for the edges. The ...
2
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1answer
180 views

what is an “edge disjoint spanning tree”?

if there are n = 2 vertices in a connected graph, i am supposed to have "n/2 edge disjoint spanning trees". This means i should have 1 edge disjoint spanning tree for a n = 2 graph? My best guess ...
2
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1answer
43 views

Graph with fixed amount of spanning trees

"Find a graph with 8 vertices, which have exactly 27 spanning trees." How do I find such a graph, or prove one does not exist?
2
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1answer
214 views

Proof involving maximum weight of edge in minimum spanning tree

Let $G$ be a minimum spanning tree of a complete graph. Let $e$ be the maximum weight edge in $G$. I'd like to proof that given any other spanning tree $G'$ of this graph, being $j$ the maximum weight ...
2
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2answers
97 views

Finding an Isolated Maximum subset of tree

Given an Oriented Tree T(V,E) with n nodes, each node have an non-negative number (the numbers are not related to nodes order). A subgroup Z of V called an Isolated if it doesn't include two nodes ...
2
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1answer
152 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 $|E(G)=m$. ...
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2answers
3k 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 ...
2
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1answer
922 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 ...
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2answers
2k 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 ...
2
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1answer
94 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 ...
2
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2answers
3k 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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4answers
4k 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 = ...
2
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1answer
328 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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2answers
60 views

How to find the number of all the possible ordered trees with n edges and k leaves?

We know that a tree with n edges have n+1 nodes.So if $|B_{n+1}|$ is the number of all possible ordered trees with n+1 nodes then its true that $C_{n+1} = |B_{n+1}|$ where $C$ is the Catalan ...
2
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1answer
62 views

Existence of infinite subsequence of trees with a special condition

For rooted trees, define $children(v)$ as the number of children of the vertex $v$. Assume two operations on rooted trees: contract an edge: choose an edge $E$, join two vertices adjacent to $E$ ...
2
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1answer
59 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) ...
2
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1answer
194 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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1answer
233 views

Number of rooted subtrees of given size in infinite d-regular tree

Currently I am reading a paper where the author states: [...] It is well-known that an infinite $D$-regular rooted tree contains precisely $\frac{1}{(D-1)u + 1} \binom{Du}{u}$ rooted subtrees of ...
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2answers
80 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 ...
2
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1answer
53 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 ...
2
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1answer
68 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.
2
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4answers
2k 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 ...
2
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1answer
121 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 ...
2
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1answer
57 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 ...
2
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1answer
321 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 ...
2
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3answers
77 views

Showing that the complete bipartite graph $K_{a,b}$ is a tree if and only if $a=1$ or $b=1$.

Let $K_{a,b}$ be the complete bipartite graph. Show that $K_{a,b}$ is a tree if and only if $a = 1$ or $b = 1$. The way my professor showed us for a complete graph is as below. I just don't know how ...
2
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1answer
37 views

Dual graph of a tree

It is stated here that: For any connected embedded planar graph G define the dual graph G* by drawing a vertex in the middle of each face of G, and connecting the vertices from two adjacent ...
2
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1answer
278 views

Longest Path in undirected unweighted graph

I came across a problem where I have to find out the longest path in a given graph. I have list of edges ( eg.{AB, BC} ) which states there is an edge between vertices/nodes (A,B,C). Now i want to ...
2
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1answer
146 views

Furthest distance vertices undirected tree

I know in my mind that it's very obvious, but I just can't seem to prove the following statement: Let $G$ be an undirected non-trivial tree with at least $3$ vertices. Let $u$ be an arbitrary vertex ...
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1answer
90 views

what' is the number of full subtrees of a full binary tree?

I'm looking for the number of full sub-trees of a binary tree; all possible tress of height less than $4$ are: Now my question is: What is $N(h)$ the maximum number of full sub-trees of a ...
2
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1answer
89 views

Number of rooted subtrees with m edges of a p-regular tree

I have the following question: Assume I have an infinite $p$-regular tree, that is a tree where every node has degree $p$ (so also the root should have degree $p$). Then how many subtrees containing ...
2
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1answer
139 views

Relax function on Bellman Ford Algorithms

In a Weighted Directed Graph $G$ (with positive weights), with $n$ vertex and $m$ edges, we want to calculate the shortest path from vertex $1$ to other vertexes. we use $1$-dimensional array $D = ...
2
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2answers
172 views

Two disjoint spanning trees, spanning subgraph with all even degrees

Show that if a graph has two edge-disjoint spanning trees then it has a connected, spanning subgraph with all degrees even. I start by looking at the union of the two spanning trees. I know it has ...
2
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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?
2
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1answer
401 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) ...