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A rooted tree T that is a subgraph of some graph G is a normal tree if the ends of every T-path in G are comparable in this tree-order (Diestel 2005, p. 15). Rooted trees, often with an additional structure such as an ordering of the neighbors at each vertex, are a key data structure in computer science; see tree data structure.
The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
Cayley's formula immediately gives the number of labelled rooted forests on n vertices, namely (n + 1) n − 1. Each labelled rooted forest can be turned into a labelled tree with one extra vertex, by adding a vertex with label n + 1 and connecting it to all roots of the trees in the forest.
The term arborescence comes from French. [6] Some authors object to it on grounds that it is cumbersome to spell. [7] There is a large number of synonyms for arborescence in graph theory, including directed rooted tree, [3] [7] out-arborescence, [8] out-tree, [9] and even branching being used to denote the same concept. [9]
The left chain of T is a sequence of ,, …, nodes such that is the root and all nodes except have one child connected to their left most (i.e., []) pointer. Any m- ary tree can be transformed to a left-chain tree using sequence of finite left-t rotations for t from 2 to m .
Trees with a single root may be viewed as rooted trees in the sense of graph theory in one of two ways: either as a tree (graph theory) or as a trivially perfect graph. In the first case, the graph is the undirected Hasse diagram of the partially ordered set, and in the second case, the graph is simply the underlying (undirected) graph of the ...
In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. [1] [2] Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions that allow multiple roots. Examples of rooted graphs with some variants.
The root of the tree is the empty sequence. In order theory , a different notion of a tree is used: an order-theoretic tree is a partially ordered set with one minimal element in which each element has a well-ordered set of predecessors.
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