Pruned tree

In descriptive set theory, a tree on a set X is a set of finite sequences of elements of X that is closed under initial segments.

More formally, it is a subset T of X^{<\omega}, such that if

\langle x_0,x_1,\ldots,x_{n-1}\rangle \in T

and 0\le m,


\langle x_0,x_1,\ldots,x_{m-1}\rangle \in T.

In particular, every nonempty tree contains the empty sequence.

A branch through T is an infinite sequence

\vec x\in X^{\omega} of elements of X

such that, for every natural number n,

\vec x|n\in T,

where \vec x|n denotes the sequence of the first n elements of \vec x. 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.

A node (that is, element) of T is terminal if there is no node of T properly extending it; that is, \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 with no terminal nodes is called pruned.

If we equip X^\omega with the product topology (treating X as a discrete space), then every closed subset of X^\omega is of the form [T] for some pruned tree T (namely, T:= \{ \vec x|n: n \in \omega, x\in X\}). Conversely, every set [T] is closed.

Frequently trees on cartesian products X\times Y are considered. In this case, by convention, the set (X\times Y)^{\omega} is identified in the natural way with a subset of X^{\omega}\times Y^{\omega}, and [T] is considered as a subset of X^{\omega}\times Y^{\omega}. 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]\}

Every tree in the sense described here is also a tree in the wider sense, i.e., the pair (T, <), where < is defined by

x<yx is a proper initial segment of y,

is a partial order in which each initial segment is well-ordered. The height of each sequence x is then its length, and hence finite.

Conversely, every partial order (T, <) where each initial segment { y: y < x0 } is well-ordered is isomorphic to a tree described here, assuming that all elements have finite height.

See also


This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.