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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317 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
268 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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94 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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58 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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2answers
62 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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2answers
94 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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1answer
132 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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30 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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2answers
39 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
313 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
72 views

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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2answers
352 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 ...
2
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1answer
43 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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1answer
93 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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3answers
665 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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56 views

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
235 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
168 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
75 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
56 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
30 views

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
135 views

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
236 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 ...
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1answer
216 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
651 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
69 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
665 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
71 views

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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0answers
120 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
241 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 $T_2$ ...
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51 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
87 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
72 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
50 views

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
158 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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47 views

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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115 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
3k 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 ...
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1answer
54 views

Expected number of feed-forward/backward triangles in a random graph with internal nodes.

Suppose we have a graph with N* nodes (these are internal nodes. they all have at least one child). Every directed link in the network exists with probability p. What would be the expected number of: ...
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2answers
74 views

Can we construct from $[0,\omega_1)$ a space which is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

I have asked in here a question which tured out to make no sense. I think I have found the confusion and would like to try and rephrase my question: Let $E$ be a topological space, $q \in E$. ...
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1answer
65 views

calculate the proportion of n-node trees whose root has only one or two subtrees.

Could we use combinatorics and generating functions to calculate the proportion of n-node trees whose root has only one or two subtrees? Here is what I tried: The combinatorial construction for the ...
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1answer
227 views

Determine number of directed trees and rooted trees obtainable

I've been doing some exercices about graph theory and I find myself stuck on this one with no idea of to proceed. Here's the question : how many different directed trees can be obtained if we assign ...
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1answer
1k views

Determining the total degree of a tree

At the start of the solution, I understand that any tree with four vertices has three edges. I don't understand the next statement: "Thus the total degree of a tree with four vertices must be 6." ...
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2answers
68 views

Finite trees and embedding in infinite regular trees.

Assume that you have a finite tree $T=(V,E)$, where $V$ and $E$ are the set of vertices and edges of $T$, respectively. Let $d_{max}$ be the maximum degree the some vertice(s) $v\in{V}$. Assume also ...
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1answer
62 views

Proving that a sub-graph of a tree is a tree

The proof that P ::== any sub-graph, G* of the tree G, is also a tree, involves proof by contradiction. We can suppose that the sub-graph has a cycle --> the whole graph has a cycle --> the whole ...
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1answer
60 views

Proving this tree definition with pigeonhole principle

I am studying the following tree definition: Let $T$ be a finite set and a function: $p: T \mathbin{\backslash} \{r\} \rightarrow T$. Then, $(T,p)$ is a tree if and only if, for all $x \in T, p^k(x) ...
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1answer
100 views

How do I construct a minimum spanning forest?

I realize that a minimum spanning forest in a weighted graph is a spanning forest with minimal weight. Does this mean that I construct it by turning all of the trees into spanning trees?
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2answers
2k views

How many trees are in the spanning forest of a graph?

Spanning forest is defined by the following definition: A forest that contains every vertex of G such that two vertices are in the same tree of the forest when there is a path in G between these two ...
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1answer
201 views

Definition of a leaf in a tree

Across two different texts, I have seen two different definitions of a leaf 1) a leaf is a node in a tree with degree 1 2) a leaf is a node in a tree with no children The problem that I see with ...