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.

learn more… | top users | synonyms

3
votes
2answers
145 views

König's Infinity Lemma and Aronszajn Trees

I am working through the notes of my Set Theory lecture. There my professor wrote: 'Is there an uncountable $\kappa$ such that K├Ânig's Infinity Lemma holds for $\kappa$? There are models where ...
2
votes
1answer
302 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: ...
9
votes
2answers
310 views

How can I prove the identity $2(n-1)n^{n-2}=\sum_k\binom{n}{k}k^{k-1}(n-k)^{n-k-1}$?

How can I prove the identity $$2(n-1)n^{n-2}=\sum_k\binom{n}{k}k^{k-1}(n-k)^{n-k-1}?$$ I know that the number of trees on $n$ vertices is $n^{n-2}$, and that a tree with $n$ vertices has $n-1$ ...
3
votes
1answer
447 views

Spanning Trees of the Complete Graph Avoiding a Given Tree

EDIT: I think everyone understood, but I never explicitly stated that I am looking at labeled spanning trees. Let $T$ be a tree contained in $K_n$ (the complete graph on $n$ vertices). How can one ...
0
votes
1answer
299 views

Explicit bijection between ordered trees with $n+1$ vertices and binary trees with $n+1$ leaves

What is an example of a direct bijection between ordered trees with $n+1$ vertices and binary trees with $n+1$ leaves?
12
votes
2answers
710 views

Partition a binary tree by removing a single edge

The question is : B-3 Bisecting trees Many divide-and-conquer algorithms that operate on graphs require that the graph be bisected into two nearly equal-sized subgraphs, which are induced by a ...
2
votes
3answers
3k views

How to show that every connected graph has a spanning tree, working from the graph “down”

I am confused about how to approach this. It says: Show that every connected graph has a spanning tree. It's possible to find a proof that starts with the graph and works "down" towards the ...
2
votes
3answers
344 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 ...
3
votes
1answer
56 views

What's the rank of this well founded relation?

Definition A tree is an ordered list of trees. (N.B these are finite objects and there is a very simple computable bijection of them with $\mathbb N$) Examples [] and [[],[],[]] and ...
0
votes
2answers
1k 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 ...
5
votes
2answers
364 views

How many vertices of degree 1 in a tree?

How many vertices of degree 1 are there in a tree with no vertices of degree more than 4? The only thing that I have right now is that the number of edges in a tree is n-1 where n is the number of ...
3
votes
2answers
198 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 ...
2
votes
1answer
1k views

Show that Minimum Spanning Tree is unique

Show that MST is unique in case the edge weights are pairwise different $(w(e)\neq w(f) \text{ for } e\neq f)$. I thought that the proof can be done for example by contradiction, saying that we ...
2
votes
2answers
1k 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 ...
1
vote
3answers
784 views

Graph with cycles proof questions

Two questions I'm stuck with: If C is a cycle, and e is an edge connecting two nonadjacent nodes of C, then we call e a chord of C. Prove that if every node of a graph G has degree at least 3, then ...
1
vote
1answer
486 views

Prove that in every tree, any two paths with maximum length have a node in common.

Prove that in every tree, any two paths with maximum length have a node in common. This is not true if we consider two maximal (i.e. non-extendable) paths. What does this even mean?
1
vote
3answers
1k views

Sufficient conditions on degrees of vertices for existence of a tree

I am answering a question for an assignment, but I am not sure if my proof is valid, can someone look at it for me? the question: "there is a tree with $p$ vertices. If $d_1, d_2, \dots , d_p$ are ...
0
votes
1answer
135 views

Number of nodes in binary tree given number of leaves

How would I prove that any binary tree that has n leaves has precisely $2n-1$ nodes ? Given that a binary tree is either a single node "o" or a node with the left and right subtrees contains a binary ...
0
votes
2answers
403 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 ...