Tree (descriptive set theory)  From Wikipedia, the free
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mathematical trees described by prefixes of finite sequences.
For trees described by partially ordered sets, see Tree (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.    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[edit]  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 th