infinitetree

Infinitetree is a theoretical construct in graph theory and computer science used to model hierarchical structures that extend without bound. It is typically described as a rooted, infinite tree: a connected acyclic graph with a distinguished root vertex such that every node except the root has a single parent, and the set of nodes is infinite.

Variants include finitely branching infinitetrees, where each node has a finite number of children, and k-ary infinitetrees where each node has exactly k children. The height may be infinite (no finite bound on depth) while breadth at each level may be finite or infinite. In the infinite setting, a path from the root to infinity is called an infinite path.

König's lemma states that every finitely branching infinite tree contains an infinite path; this result is used in logic and combinatorics. Related concepts include Aronszajn trees, which have height ω1 and no branches of length ω1, illustrating diversity among infinite trees.

In applications, infinitetrees serve as abstract models in automata theory, model checking, and the study of infinite derivations and hierarchies. They also appear in set theory and descriptive set theory as objects of study for branches, ranks, and well-foundedness. In practical computation, infinite trees are represented procedurally or lazily, using generators, coinduction, or guarded recursion, allowing analysis without materializing the entire structure.

See also: Tree (graph theory); Infinite tree; König's lemma; Aronszajn tree; Automata theory; Model checking.