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Tree (graph theory)

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Trees

A labeled tree with 6 vertices and 5 edges
Vertices v
Edges v - 1
Chromatic number 2
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In mathematics, more specifically graph theory, a tree is a graph in which any two vertices are connected by exactly one path. Alternatively, any connected graph with no cycles is a tree. A forest is a disjoint union of trees. Trees are widely used in computer science data structures such as binary search trees, heaps, tries, Huffman trees for data compression, etc.

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[edit] Definitions

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

  • G is connected and has no cycles.
  • G has no cycles, and a simple cycle is formed if any edge is added to G.
  • G is connected, and it is not connected anymore if any edge is removed from G.
  • G is connected and the 3-vertex complete graph K3 is not a minor of G.
  • 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 any of the following conditions:

  • 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 directed tree is a directed graph which would be a tree if the directions on the edges were ignored. Some authors restrict the phrase to the case where the edges are all directed towards a particular vertex, or all directed away from a particular vertex (see arborescence.)

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. The tree-order is the partial ordering on the vertices of a tree with u ≤ v if and only if the unique path from the root to v passes through u. A tree which is a subgraph of some graph G is a normal tree if the ends of every edge in G are comparable in this tree-order (Diestel 2005, p. 15). 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. In a context where trees are supposed to have a root, a tree without any designated root is called a free tree.

A polytree has at most one undirected path between any two vertices. In other words, a polytree is a directed acyclic graph (DAG) for which there are no undirected cycles either.

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. A recursive tree is a labeled rooted tree where the vertex labels respect the tree order (i.e., if u < v for two vertices u and v, then the label of u is smaller than the label of v).

An irreducible (or series-reduced) tree is a tree in which there is no vertex of degree 2.

An ordered tree is a tree for which an ordering is specified for the children of each node.

An n-ary tree is a tree for which each node which is not a leaf has at most n children. 2-ary trees (resp. 3-ary trees) are sometimes called binary trees (resp. ternary trees)

[edit] 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.

[edit] Facts

  • 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. Every connected graph even admits a normal spanning tree (Diestel 2005, Prop. 1.5.6).
  • Every finite tree with at least two vertices, say n, has at least two leaves or vertices of degree 1. The minimal number of leaves corresponds to the path graph and the maximal number (n - 1) corresponds to the star graph.
  • For any three vertices in a tree, the three paths between them have at least one vertex in common.

[edit] Enumeration

Given n labeled vertices, there are nn−2 different ways to connect them to make a tree. This result is called Cayley's formula. It can be proved by first showing that the number of trees with n vertices of degree d1,d2,...,dn is the multinomial coefficient

{n-2 \choose d_1-1, d_2-1, \ldots, d_n-1}.

Counting the number of unlabeled trees is a harder problem. No closed formula for the number t(n) of trees with n vertices up to graph isomorphism is known. Otter (1948) proved that

t(n) \sim C \alpha^n n^{-5/2} \quad\text{as } n\to\infty,

with C = 0.53495… and α = 2.95576… (here, f \sim g means that \lim_{n \to \infty} f/g = 1).

[edit] Types of trees

See List of graph theory topics: Trees.

[edit] See also

[edit] References

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