Tree (descriptive set theory)

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

Trees
The collection of all finite sequences of elements of a set $$X$$ is denoted $$X^{<\omega}$$. With this notation, a tree is a nonempty subset $$T$$ of $$X^{<\omega}$$, such that if $$\langle x_0,x_1,\ldots,x_{n-1}\rangle$$ is a sequence of length $$n$$ in $$T$$, and if $$0\le m<n$$, then the shortened sequence $$\langle x_0,x_1,\ldots,x_{m-1}\rangle$$ also belongs to $$T$$. In particular, choosing $$m=0$$ shows that the empty sequence belongs to every tree.

Branches and bodies
A branch through a tree $$T$$ is an infinite sequence of elements of $$X$$, each of whose finite prefixes belongs to $$T$$. The set of all branches through $$T$$ is denoted $$[T]$$ and called the body of the tree $$T$$.

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 $$T$$ is called a terminal node if it is not a prefix of a longer sequence in $$T$$. Equivalently, $$\langle x_0,x_1,\ldots,x_{n-1}\rangle \in T$$ is terminal if there is no element $$x$$ of $$X$$ such that that $$\langle x_0,x_1,\ldots,x_{n-1},x\rangle \in T$$. 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 $$T$$ is a tree in the descriptive set theory sense, then it corresponds to a graph with one vertex for each sequence in $$T$$, 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 $$T$$ and $$U$$ are ordered by $$T<U$$ if and only if $$T$$ is a proper prefix of $$U$$. 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).

Topology
The set of infinite sequences over $$X$$ (denoted as $$X^\omega$$) may be given the product topology, treating X as a discrete space. In this topology, every closed subset $$C$$ of $$X^\omega$$ is of the form $$[T]$$ for some pruned tree $$T$$. Namely, let $$T$$ consist of the set of finite prefixes of the infinite sequences in $$C$$. Conversely, the body $$[T]$$ of every tree $$T$$ forms a closed set in this topology.

Frequently trees on Cartesian products $$X\times Y$$ are considered. In this case, by convention, we consider only the subset $$T$$ of the product space, $$(X\times Y)^{<\omega}$$, containing only sequences whose even elements come from $$X$$ and odd elements come from $$Y$$ (e.g., $$\langle x_0,y_1,x_2,y_3\ldots,x_{2m}, y_{2m+1}\rangle$$). Elements in this subspace are identified in the natural way with a subset of the product of two spaces of sequences, $$X^{<\omega}\times Y^{<\omega}$$ (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 $$[X^{<\omega}]\times [Y^{<\omega}]$$ with $$[T]$$ for over the product space. We may then form the projection of $$[T]$$,
 * $$p[T]=\{\vec x\in X^{\omega} | (\exists \vec y\in Y^{\omega})\langle \vec x,\vec y\rangle \in [T]\}$$.