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In descriptive set theory, a **tree** on a set is a collection of finite sequences of elements of such that every prefix of a sequence in the collection also belongs to the collection.

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.

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

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**.

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 (the set of all of its prefixes). An order-theoretic tree may be represented by an isomorphic tree of sequences if and only if each of its elements has finite height (that is, a finite set of predecessors).

The set of infinite sequences over (denoted as ) may be given the product topology, treating *X* 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 (e.g., ). Elements in this subspace are identified in the natural way with a subset of the product of two spaces of sequences, (the subset for which the length of the first sequence is equal to or 1 more than the length of the second sequence).
In this way we may identify with for over the product space. We may then form the **projection** of ,

- .

- Laver tree, a type of tree used in set theory as part of a notion of forcing

- Kechris, Alexander S. (1995).
*Classical Descriptive Set Theory*. Graduate Texts in Mathematics 156. Springer. ISBN 0-387-94374-9 ISBN 3-540-94374-9.