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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Proof about spanning tress in graphs

Let $G=(V,E)$ be a graph and $T_i=(V,F_i),i=1,2$ two disjoint spanning trees in $G$. Let $f_1 \in F_1$. Prove that there is $f_2\in F_2 $ such that $T:=T_1-f_1+f_2$ is a spanning tree.
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Finding number of homeomorphically irreducible trees of degree N

There is a scene in Goodwill Hunting where professor challenges students with task of finding all homeomorphically irreducible trees of degree 10. This is discussed in many places, such as here and is ...
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How does inserting N objects one at a time into an ordered AVL tree yield an efficient sorting algorithim

If we assume reblalancing an AVL tree of height n after an insertion or deletion takes O(n) operations. How does inserting N objects one at a time into an ordered AVL tree yield an efficient sorting ...
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2answers
37 views

Prove that if G is a tree in which all vertices have odd degree then G has odd size.

Prove that if G is a tree in which all vertices have odd degree then G has odd size. Good night, do not know how to approach this "prove". Can you give me tips to solve it?. Please.
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1answer
19 views

Formula for number of “root” nodes in a tree where Parent shares child nodes?

If I have a tree like this: {a},{b,c},{d,e,f},{g,h,i,j} in this case we have a total of 10 nodes. Is there any equation where given "10" I can calculate how many bottom nodes there are (answer: "4" ...
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3answers
71 views

Proof of an $\iff$ statement on binary trees

Let $x$ and $y$ be two nodes of a binary tree $B$. Prove that $x$ is an ancestor of $y$ $\iff$ $x$ stands before $y$ in the pre-order traversal of $B$ and $x$ stands after $y$ in the ...
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1answer
30 views

How to compute a marginal probability

Given a weighted graph, using the Kirchhoff's matrix tree theorem, how can I compute the marginal edge presence probability: $$P_\beta(ij)=Z_\beta^{-1}\sum_{\text{T spanning tree:$(i,j)\in ...
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49 views

Why is the height of a heap defined as $\lg n$?

I'm a bit confused about why the height of a heap (or a binary tree in general) is given by the floor of $\lg n$. E.g. if you have a tree with 7 nodes, you would get $h = 0$ instead of $h = 2$. Isn't ...
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35 views

Min. Spanning Tree - Same weight

Prove that every minimum spanning tree of a connected graph, $G$, has the same maximum edge. Intuitively, this makes sense to me. You need to have that heavy edge because that is the cheapest ...
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2answers
40 views

Tree with $k$ edges is a subgraph of any graph with all vertices of degree $\geq k$.

Let $T$ be a tree with $k$ edges. Let $G$ be a graph where every vertex has a degree of at least $k$. Show that $T$ is a subgraph $G$. I know this implies that in a graph where every vertex is at ...
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49 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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1answer
22 views

Generators Trees in a Tree

My question is very short: How many spanning trees have a tree? Thanks in advance
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1answer
242 views

Number of binary trees with N nodes

I am trying to calculate the number of trees (non isomorphic) with n nodes (total including leaves). I think that there are n! such trees, but I don't know how to prove that. I know that the number ...
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2answers
31 views

Number of spanning trees of a graph (behind the formula)

Given $G$ a subgraph of $K_n$ s.t. $G$ has $n$ vertices with adjacency matrix $A$; why is $$\sum_{T \text{ spanning tree of }K_n}\prod_{(i,j)\in T}A_{i,j}$$ the number of spanning trees? I can't get ...
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1answer
119 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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0answers
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Proof on Dyadic Trees [Smullyan: First-Order Logic, chapter 1, section 0]

I'm having difficult with a proof from Smullyan's First-Order Logic, Chapter 1 Section 0 (Reprint, Dover 1968, p. 4): Prove: In a dyadic tree, define x to be to the left of y if there is a ...
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271 views

Does every “balloon” (dragon, tadpole, canoe paddle) admit a graceful labeling?

Post-Bounty Edit: Still no answer; will gladly accept if someone can provide a reference. Earlier Edit: It appears that the answer is "yes," either by an already existent publication or by ...
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2answers
107 views

Number of all labeled, unordered rooted trees with $n$ vertices and $k$ leaves.

I've been trying to do the following exercise: The problem Find the number of all labeled, unordered rooted trees with $n$ vertices and $k$ leaves. I know that I should try to write an ...
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0answers
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Find tree diameter or center

I want to find center in a graph that doesn't have cycles. I heard, that this is how I find a diameter: Take random vertex A Find such vertex B, that distance to it is maximal Find such vertex C, ...
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24 views

Can a non-cyclic infinite proof tree with always-reachable provable nodes be used to construct a proof?

Suppose that I have a finite number of basic elements x,y,z ... and a finite number of operators +, * ... Terms X,Y,Z ... are created by combining basic elements and operators. For example, x+y, and ...
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18 views

$k$ edge-disjoint $r$-arborescences in an acylic digraph

An $r$-arborescence of a digraph $D$ is a rooted spanning tree with root $r\in V(D)$ in which all the edges of $D$ are directed away from $r$. I would like to prove the following: I have thought ...
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1answer
33 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 ...
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Free medial magmas

A medial magma is a set $M$ with a binary relation $*$ satisfying $(a*b)*(c*d) = (a*c)*(b*d)$. Medial magmas constitute an algebraic category $\mathsf{Med}$, therefore there is a functor $\mathsf{Set} ...
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1answer
54 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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307 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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maximizing the inverse degree in a graph

The inverse degree in the graph $G$ is defined as \begin{align*} r(G) = \sum_{i=1}^N \frac{1}{d_i}, \end{align*} where $d_i$ is the degree of node (vertex) $i$. Is the connected graph with maximum ...
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2answers
207 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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2answers
142 views

An identity on the number of trees

Let $T_n$ be the number of labelled trees on $n$ vertices, then $$ T_n=\sum_kk\binom{n-2}{k-1}T_kT_{n-k} \tag{1}$$ Using this question, I was able to prove that $$ T_n= \frac{n}{2} \ ...
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1answer
23 views

Prove that every subgraph of forest has at least one vertex of degree < 2

So I know that a forest is a graph that has no cycles. This is what I had in mind: Assume that we have the subgraph T, which has two options: to be connected or not. if it's connected it has to be a ...
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2answers
21 views

Determine the minimum number of weighings to find the counterfeit coin.

Here's the full problem: We have 20 coins, 1 of which is counterfeit (too light). Determine the minimum number of weighings to find the counterfeit coin. Okay so is used the formula $$h=\left ...
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1answer
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Finding the number of spanning trees on a set of vertices.

I need to find the number of spanning trees on $V = \{1,2,3,4,5,6,7,8,9\}$, where $\{1,2,3,4\}$ are leaves. Can anyone tell me how?
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3answers
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is MST a Steiner tree?

I am a little bit confused about MST and Steiner tree? Is an MST a steiner tree?? and suppose we are given a weighted undirected connected graph G = (V,E) and S ⊆ V is the smallest subtree of an MST ...
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2answers
165 views

Generating function for planted planar trees

I need your help to solve this problem : Give a generating function for planted planar trees with all degrees odd. Show that the number of such trees with $2k+1$ non-root vertices is ...
2
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1answer
163 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
376 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 ...
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2answers
53 views

When trees are the answer: what is the question?

For which optimization problems are (abstract) trees the best solution? E.g. binary search trees are somehow optimal data structures for quick search. But why for example do botanic trees grow as ...
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1answer
47 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 ...
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1answer
47 views

Proof for binary tree is a planar graph

Suppose G is a binary tree. Is G necessarily planar? Give a proof, or a counterexample. My guess is that it is indeed planar but I am struggling to find a formal proof for this. EDIT: Is there a ...
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2answers
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Spanning Tree, Network Modelling

I'm developing some software at the moment for voip communications (broadcast style comms, think ventrilo or teamspeak) between multiple users without a central server (send voice to server, server ...
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1answer
722 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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19 views

A function for the branches of a tree

Imagine that we have to calculate the number of branches of a tree. Now I made a formula that associates the "level" (number of the times that a branch replicates itself) $k$ of the tree and the ...
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1answer
163 views

Proofs involving subtrees of a tree

I have found some claims about trees in my graph theory text, and I am wondering if corresponding proofs can be found, as I cannot find any online or in another text. First, If $T_1$ and $T_2$ ...
0
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1answer
26 views

Proof that a local minimum in a spanning tree is also a minimum spanning tree.

Be $G$ a connected graph with weights associated to its edges. Be $T(G)$ the graph that has the spanning trees of $G$ as vertex, and two spanning trees are adjacent to each other if and only if each ...
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2answers
64 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
28 views

Distinct MST Edge Proof

Suppose that T1 and T2 are distinct MSTs for an undirected graph G. Let (u,v) be the lightest edge that is in T2 and not in T1. Let (x,y) be any edge that is in T1 and not in T2. What can you say ...
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1answer
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Find MST based on new definition

Redefine the weight of a spanning tree to be the weight of the maximum weight edge in the tree (i.e. the weight of the tree is no longer the sum of the weights of all the edges in the tree, only the ...
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2answers
44 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 ...
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1answer
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How many spanning trees of a complete graph with an even number of vertices can be split in half by removing a single edge?

We have a complete Graph G with |V|=n . We know it has n^(n-2) possible spanning trees. How many of them could be split into two equal halves by removing a single edge?
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33 views

heavy path decomposition/heavy-light decomposition

I'm reading about the heavy path decomposition of trees and its application. I would like to know its time and space complexity. In addition, can the algorithm be implemented in distributed way? What ...
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1answer
949 views

Height of a full binary tree

A full binary tree seems to be a binary tree in which every node is either a leaf or has 2 children. I have been trying to prove that its height is O(logn) unsuccessfully. Here is my work so far: I ...