Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [trees]

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.

Trees are an important topic in graph theory. A tree is a connected graph with no cycles. They are of enormous practical importance in applied mathematics.

For example, in a connected weighted graph, it is interesting to know how to find a subgraph which is a tree and which maximizes or minimizes the total weight of the subgraph. A subgraph of a graph $G$ which is a tree that touches all nodes of $G$ is called a spanning tree for $G$.

For instance, imagine a network of electrical systems, such as a telephone system, between nodes. Suppose that signals can travel through any path in the network, passing through intervening nodes to get to the final destination. The amount of wiring can be reduced by deleting portions of cycles in the graph of the network. The graph can eventually be reduced to a tree, and all nodes remain connected and accessible through the network.

Dijkstra's algorithm is an effective way to determine the lowest-weight spanning tree for a given connected graph.

2113 questions
75
votes
1 answer

Why is TREE(3) so big? (Explanation for beginners)

I am not a mathematician but I am interested in big numbers. I find them to be really interesting, almost god-like. I am watching a series of videos from David Metzler on YouTube. I have a basic understanding of some fast growing functions. David…
Josh Kerr
  • 957
49
votes
5 answers

Show that there's a unique minimum spanning tree if all edges have different costs

Show that there's a unique minimum spanning tree (MST) in case the edges' 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 have $2$ different…
totpiko
  • 647
28
votes
7 answers

Finding the number of Spanning Trees of a Graph $G$

This is my first question on the Mathematics StackExchange site, so please do tell me if I have not adhered to any conventions in this post. Ok, so if I have a graph $G$, with say $6$ vertices and $7$ Edges, how would I determine how many possible…
Miles
  • 412
28
votes
1 answer

Proof that TREE(n) where n >= 3 is finite?

Reading online, it generally seems accepted that TREE(n) where n >= 3 is a finite number, but large enough to be incomputable and only has extremely loose lower bounds today. TREE(n) is the function defined by Harvey Friedman, based on Joseph…
Ivan G.
  • 403
23
votes
4 answers

Determining Ambiguity in Context Free Grammars

What are some common ways to determine if a grammar is ambiguous or not? What are some common attributes that ambiguous grammars have? For example, consider the following Grammar G: $S \rightarrow S(E)|E$ $E \rightarrow (S)E|0|1|\epsilon$ My guess…
CodeKingPlusPlus
  • 7,538
23
votes
5 answers

The maximum number of nodes in a binary tree of depth $k$ is $2^{k}-1$, $k \geq1$.

I am confused with this statement The maximum number of nodes in a binary tree of depth $k$ is $2^k-1$, $k \geq1$. How come this is true. Lets say I have the following tree 1 / \ 2 3 Here the depth of the tree is 1. So according to the…
rajan sthapit
  • 459
21
votes
0 answers

Free medial magmas

A medial magma is a set $M$ with a binary operation $*$ satisfying $$(a*b)*(c*d) = (a*c)*(b*d)$$ for all $a,b,c,d \in M$. Medial magmas constitute a finitary algebraic category $\mathsf{Med}$, therefore there is a functor $M : \mathsf{Set} \to…
Martin Brandenburg
  • 181,922
20
votes
4 answers

Proving every tree has at most one perfect matching

In trying to prove that every tree, T, has at most one perfect matching, I came across this idea: Since the matchings are perfect, each vertex has degree $0$ or $2$ in the symmetric difference, so every component is an isolated vertex or a…
larry
  • 1,526
19
votes
0 answers

Is the Frog game solvable in the root of a full binary tree?

Frog game The Frog game is the generalization of the Frog Jumping (see it on Numberphile) that can be played on any graph, but by convention, we restrict the game to Tree graphs (see wikipedia). The game is simple to play, but it can be hard to…
Vepir
  • 13,072
18
votes
2 answers

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

8/18/14 Edit: If anyone has a copy of a related reference, then I would be happy to see it. For now, I am accepting the answer below and considering the question answered in the affirmative: Yes. Earlier Edit: It appears that the answer is "yes,"…
Benjamin Dickman
  • 16,038
18
votes
5 answers

How many edges does an undirected tree with $n$ nodes have?

How many edges does an undirected tree with $n$ nodes have?
siamii
  • 297
16
votes
2 answers

Is there a "tree-like" proof of compactness theorem in the case of uncountably many variables?

I like proofs using trees and König's lemma, since they are very visual. One of the applications of König's lemma you can show to students is proving compactness theorem for propositional calculus, which says that a set of formula is satisfiable if…
Martin Sleziak
  • 56,060
15
votes
5 answers

Difference between a tree and spanning tree?!

I'm unable to understand the difference between a tree and a spanning tree. A tree is a graph that is connected and contains no circuits. A spanning tree of a graph G is a tree that contains every node of G. So what is the difference!?!
Jack Twain
  • 1,234
  • 5
  • 17
  • 32
15
votes
1 answer

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 partition of the vertices. This problem…
LeafGlowPath
  • 7,683
15
votes
6 answers

Prove that in a tree with maximum degree $k$, there are at least $k$ leaves

Prove that in a tree with a maximum degree for each vertex is $k$, there are at least $k$ leaves. So I said: $2|E| = \sum_{v \in V} {\deg(v)} \leq k $ which is, if we say that we have AT MOST $k-1$ leaves (I used the contradiction method to…
TheNotMe
  • 4,899
1
2 3
…
99 100