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Tree (graph theory)#Rooted tree
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Rooted tree
A rooted tree is a term used in graph theory to describe a particular kind of undirected graph. It's a tree that has been distinguished by one of its vertices, which is then designated as the root. This designation fundamentally alters how we perceive and analyze the structure. Imagine a regular tree, perhaps one you'd find in a forest, with its branches spreading out. Now, pick one specific point on that tree – maybe the base of the trunk, or a peculiar knot – and declare that to be the "root." Everything else then relates to this chosen point.
The concept of a rooted tree is ubiquitous across various fields, from computer science to phylogenetics. In computer science, for instance, data structures like binary trees and abstract syntax trees are fundamentally rooted trees. They provide an organized way to represent hierarchical relationships, much like a family tree or an organizational chart. In phylogenetics, rooted trees are used to depict the evolutionary relationships between different species, with the root representing the common ancestor.
The distinction of a root imposes a hierarchy on the tree. Each vertex, other than the root, has exactly one parent vertex – the vertex directly connected to it on the path back to the root. This creates a clear lineage. Conversely, a vertex can have multiple child vertices, which are the vertices directly connected to it and further away from the root. Vertices with no children are called leaf vertices or terminal nodes. All other vertices are referred to as internal vertices.
The path from the root to any vertex is unique. This property is crucial for many algorithms and analyses performed on rooted trees. The depth of a vertex is defined as the number of edges on the path from the root to that vertex. The root itself has a depth of zero. The height of a vertex is the number of edges on the longest downward path from that vertex to a leaf. The height of the entire tree is the height of its root.
A rooted tree can be thought of as an arborescence, which is a directed graph where every vertex has exactly one incoming edge, except for the root, which has none. This directed perspective emphasizes the flow from the root downwards.
The definition of a rooted tree is quite precise. It is a connected, acyclic graph where one vertex has been designated as the root. This initial designation imbues the entire structure with a sense of directionality and hierarchy that wouldn't exist in an unrooted tree. It’s like giving a map a “You Are Here” marker – suddenly, all other locations are understood in relation to that point. The absence of cycles, a defining characteristic of any tree, ensures that there are no loops or redundant paths, making the structure efficient and unambiguous for navigation and analysis.