Definition in Graph Theory
For each binary tree data structure, there is equivalent rooted binary tree in graph theory.
Graph theorists use the following definition: A binary tree is a connected acyclic graph such that the degree of each vertex is no more than three. It can be shown that in any binary tree of two or more nodes, there are exactly two more nodes of degree one than there are of degree three, but there can be any number of nodes of degree two. A rooted binary tree is such a graph that has one of its vertices of degree no more than two singled out as the root.
With the root thus chosen, each vertex will have a uniquely defined parent, and up to two children; however, so far there is insufficient information to distinguish a left or right child. If we drop the connectedness requirement, allowing multiple connected components in the graph, we call such a structure a forest.
Another way of defining binary trees is a recursive definition on directed graphs. A binary tree is either:
- A single vertex.
- A graph formed by taking two binary trees, adding a vertex, and adding an edge directed from the new vertex to the root of each binary tree.
This also does not establish the order of children, but does fix a specific root node.
Read more about this topic: Binary Tree
Famous quotes containing the words definition, graph and/or theory:
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“In this Journal, my pen is a delicate needle point, tracing out a graph of temperament so as to show its daily fluctuations: grave and gay, up and down, lamentation and revelry, self-love and self-disgust. You get here all my thoughts and opinions, always irresponsible and often contradictory or mutually exclusive, all my moods and vapours, all the varying reactions to environment of this jelly which is I.”
—W.N.P. Barbellion (1889–1919)
“every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.”
—Thomas Nagel (b. 1938)