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In set theory, a tree is a partially ordered set (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation <. Frequently trees are assumed to have only one root (i.e. minimal element), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.
Every tree in descriptive set theory is also an order-theoretic tree, using a partial ordering in which two sequences and are ordered by < if and only if is a proper prefix of . The empty sequence is the unique minimal element, and each element has a finite and well-ordered set of predecessors (the set of all of its prefixes).
In theoretical computer science and formal language theory, a regular tree grammar is a formal grammar that describes a set of directed trees, or terms. [1] A regular word grammar can be seen as a special kind of regular tree grammar, describing a set of single-path trees.
In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be connected to exactly one parent, [ 1 ] [ 2 ] except for the root node, which has no parent (i.e., the ...
For example, each node of the tree is a word over set of natural numbers (), which helps this definition to be used in automata theory. A tree is a set T ⊆ * such that if t.c ∈ T, with t ∈ * and c ∈ , then t ∈ T and t.c 1 ∈ T for all 0 ≤ c 1 < c.
A recursive definition using set theory is that a binary tree is a tuple (L, S, R), where L and R are binary trees or the empty set and S is a singleton set containing the root. [1] [2] From a graph theory perspective, binary trees as defined here are arborescences. [3]
Trees (graph theory) (2 C, 40 P) Pages in category "Trees (set theory)" The following 13 pages are in this category, out of 13 total.
P is the set of all perfect trees contained in the set of finite {0, 1} sequences. (A tree T is a set of finite sequences containing all initial segments of its members, and is called perfect if for any element t of T there is a segment s extending t so that both s0 and s1 are in T.) A tree p is stronger than q if p is contained in q.