Tree (graph theory)

From Academic Kids

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A labeled tree with 6 vertices and 5 edges

In graph theory, a tree is a graph in which any two vertices are connected by exactly one path. A forest is a graph in which any two vertices are connected by at most one path. An equivalent definition is that a forest is a disjoint union of trees (hence the name).

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Definitions

A tree is an undirected simple graph G that satisfies any of the following equivalent conditions:

G is connected and has no simple cycles
G has no simple cycles and, if any edge is added to G, then a simple cycle is formed
G is connected and, if any edge is removed from G, then it is not connected anymore
Any two vertices in G can be connected by a unique simple path.

If G has finitely many vertices, say n of them, then the above statements are also equivalent to:

G is connected and has n − 1 edges
G has no simple cycles and has n − 1 edges

An undirected simple graph G is called a forest if it has no simple cycles.

A tree is called a rooted tree if one vertex has been designated the root, in which case the edges have a natural orientation, towards or away from the root. Rooted trees, often with additional structure such as ordering of the neighbors at each vertex, are a key data structure in computer science; see tree data structure.

A labeled tree is a tree in which each vertex is given a unique label. The vertices of a labeled tree on n vertices are typically given the labels {1, 2, ..., n}.

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Example

The example tree shown to the right has 6 vertices and 6 − 1 = 5 edges. The unique simple path connecting the vertices 2 and 6 is 2-4-5-6.

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Facts

Every tree is a planar and bipartite graph.

Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are edges of G.

Given n labeled vertices, there are nⁿ⁻² different ways to connect them to make a tree. This result is called Cayley's formula.

The number of trees with n vertices of degree d₁,d₂,...,d_n is

<math> {n-2 \choose d_1-1, d_2-1, \ldots, d_n-1},<math>

which is a multinomial coefficient.

No closed formula for the number t(n) of trees with n vertices up to graph isomorphism is known. However, the asymptotic behavior of t(n) is known: there are numbers α ≈ 3 and β ≈ 0.5 such that

<math>\lim_{n\to\infty} \frac{t(n)}{\beta \alpha^n n^{-5/2}} = 1.<math>

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