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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.
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.
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).
Representations might also be more complicated, for example using indexes or ancestor lists for performance. Trees as used in computing are similar to but can be different from mathematical constructs of trees in graph theory , trees in set theory , and trees in descriptive set theory .
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.
For a given tree-language L, a congruence can be defined by u ≡ L v if C[u] ∈ L ⇔ C[v] ∈ L for each context C. The Myhill–Nerode theorem for tree automata states that the following three statements are equivalent: [14] L is a recognizable tree language; L is the union of some equivalence classes of a congruence of finite index