The collection of all finite sequences of elements of a set is denoted. With this notation, a tree is a nonempty subset of, such that if is a sequence of length in, and if, then the shortened sequence also belongs to. In particular, choosing shows that the empty sequence belongs to every tree.
Branches and bodies
A branch through a tree is an infinite sequence of elements of, each of whose finite prefixes belongs to. The set of all branches through is denoted and called the body of the tree. A tree that has no branches is called wellfounded; a tree with at least one branch is illfounded. By Kőnig's lemma, a tree on a finite set with an infinite number of sequences must necessarily be illfounded.
Terminal nodes
A finite sequence that belongs to a tree is called a terminal node if it is not a prefix of a longer sequence in. Equivalently, is terminal if there is no element of such that that. A tree that does not have any terminal nodes is called pruned.
Relation to other types of trees
In graph theory, a rooted tree is a directed graph in which every vertex except for a special root vertex has exactly one outgoing edge, and in which the path formed by following these edges from any vertex eventually leads to the root vertex. If is a tree in the descriptive set theory sense, then it corresponds to a graph with one vertex for each sequence in, and an outgoing edge from each nonempty sequence that connects it to the shorter sequence formed by removing its last element. This graph is a tree in the graph-theoretic sense. The root of the tree is the empty sequence. In order theory, a different notion of a tree is used: an order-theoretic tree is a partially ordered set with one minimal element in which each element has a well-ordered set of predecessors. 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. An order-theoretic tree may be represented by an isomorphic tree of sequences if and only if each of its elements has finite height.
Topology
The set of infinite sequences over may be given the product topology, treatingX as a discrete space. In this topology, every closed subset of is of the form for some pruned tree. Namely, let consist of the set of finite prefixes of the infinite sequences in. Conversely, the body of every tree forms a closed set in this topology. Frequently trees on Cartesian products are considered. In this case, by convention, we consider only the subset of the product space,, containing only sequences whose even elements come from and odd elements come from . Elements in this subspace are identified in the natural way with a subset of the product of two spaces of sequences, . In this way we may identify with for over the product space. We may then form the projection of,
