Tree (graph theory)

Tree

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

G is connected and has no cycles.

G is acyclic, and a simple cycle is formed if any edge is added to G.

G is connected, but is not connected if any single 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.

As elsewhere in graph theory, the order-zero graph (graph with no vertices) is generally excluded from consideration: while it is vacuously connected as a graph (any two vertices can be connected by a path), it is not 0-connected (or even (−1)-connected) in algebraic topology, unlike non-empty trees, and violates the "one more vertex than edges" relation.

An internal vertex (or inner vertex or branch vertex) is a vertex of degree at least 2. Similarly, an external vertex (or outer vertex, terminal vertex or leaf) is a vertex of degree 1.

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

Forest

A forest is an undirected graph, all of whose connected components are trees; in other words, the graph consists of a disjoint union of trees. Equivalently, a forest is an undirected acyclic graph. As special cases, an empty graph, a single tree, and the discrete graph on a set of vertices (that is, the graph with these vertices that has no edges), are examples of forests.

Polytree

A polytree (or oriented tree or singly connected network) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic.

A directed tree is a directed graph which would be a tree if the directions on the edges were ignored, i.e. a polytree. 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).

Rooted tree

A rooted tree is a tree in which one vertex has been designated the root. The edges of a rooted tree can be assigned a natural orientation, either away from or towards the root, in which case the structure becomes a directed rooted tree. When a directed rooted tree has an orientation away from the root, it is called an arborescence, branching, or out-tree; when it has an orientation towards the root, it is called an anti-arborescence or in-tree. 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 rooted 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 whenever those ends are vertices of the tree (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 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).

In a rooted tree, the parent of a vertex is the vertex connected to it on the path to the root; every vertex except the root has a unique parent. A child of a vertex v is a vertex of which v is the parent. A descendant of any vertex v is any vertex which is either the child of v or is (recursively) the descendant of any of the children of v. A sibling to a vertex v is any other vertex on the tree which has the same parent as v. The root is an external vertex if it has precisely one child. A leaf is different from the root.

The height of a vertex in a rooted tree is the length of the longest downward path to a leaf from that vertex. The height of the tree is the height of the root. The depth of a vertex is the length of the path to its root (root path). This is commonly needed in the manipulation of the various self-balancing trees, AVL trees in particular. The root has depth zero, leaves have height zero, and a tree with only a single vertex (hence both a root and leaf) has depth and height zero. Conventionally, an empty tree (tree with no vertices, if such are allowed) has depth and height −1.

A k-ary tree is a rooted tree in which each vertex has at most k children. 2-ary trees are often called binary trees, while 3-ary trees are sometimes called ternary trees.

Ordered tree

An ordered tree (or plane tree) is a rooted tree in which an ordering is specified for the children of each vertex. This is called a "plane tree" because an ordering of the children is equivalent to an embedding of the tree in the plane, with the root at the top and the children of each vertex lower than that vertex. Given an embedding of a rooted tree in the plane, if one fixes a direction of children, say left to right, then an embedding gives an ordering of the children. Conversely, given an ordered tree, and conventionally drawing the root at the top, then the child vertices in an ordered tree can be drawn left-to-right, yielding an essentially unique planar embedding .

Facts

Every tree is a bipartite graph and a median graph. Every tree with only countably many vertices is a planar 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.

Every connected graph with only countably many vertices admits a normal spanning tree (Diestel 2005, Prop. 8.2.4).

There exist connected graphs with uncountably many vertices which do not admit a normal spanning tree (Diestel 2005, Prop. 8.5.2).

Every finite tree with n vertices, with n > 1, has at least two terminal vertices (leaves). This minimal number of leaves is characteristic of path graphs; the maximal number, n − 1, is attained only by star graphs. The number of leaves is at least the maximal vertex degree.

For any three vertices in a tree, the three paths between them have exactly one vertex in common.

Otter showed that any n-vertex tree has either a unique center vertex, whose removal splits the tree into subtrees of fewer than n/2 vertices, or a unique center edge, whose removal splits the tree into two subtrees of exactly n/2 vertices.

Labeled trees

Cayley's formula states that there are nⁿ⁻² trees on n labeled vertices. A classic proof uses Prüfer sequences, which naturally show a stronger result: the number of trees with vertices 1, 2, …, n of degrees d₁, d₂, …, d_n respectively, is the multinomial coefficient

( n − 2 d 1 − 1 , d 2 − 1 , … , d n − 1 ) .

A more general problem is to count spanning trees in an undirected graph, which is addressed by the matrix tree theorem. (Cayley's formula is the special case of spanning trees in a complete graph.) The similar problem of counting all the subtrees regardless of size has been shown to be #P-complete in the general case (Jerrum (1994)).

Unlabeled trees

Counting the number of unlabeled free trees is a harder problem. No closed formula for the number t(n) of trees with n vertices up to graph isomorphism is known. The first few values of t(n) are

1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, … (sequence A000055 in the OEIS).

Otter (1948) proved the asymptotic estimate

t ( n ) ∼ C α n n − 5 / 2 as  n → ∞ ,

with the values C and α known to be approximately 0.534949606… and 2.95576528565… (sequence A051491 in the OEIS), respectively. (Here, f ~ g means that lim_n→∞ f /g = 1.) This is a consequence of his asymptotic estimate for the number r(n) of unlabeled rooted trees with n vertices:

r ( n ) ∼ D α n n − 3 / 2 as  n → ∞ ,

with D around 0.43992401257… and the same α as above (cf. Knuth (1997), chap. 2.3.4.4 and Flajolet & Sedgewick (2009), chap. VII.5, p. 475).

The first few values of r(n) are

1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32973, …

Types of trees

A star tree is a tree which consists of a single internal vertex (and n − 1 leaves). In other words, a star tree of order n is a tree of order n with as many leaves as possible.

A caterpillar tree is a tree in which all vertices are within distance 1 of a central path subgraph.

A lobster tree is a tree in which all vertices are within distance 2 of a central path subgraph.