In set theory, a tree is a partially ordered set such that for each t ∈ T, the set is well-ordered by the relation <. Frequently trees are assumed to have only one root, as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.
Definition
A tree is a partially ordered set such that for each t ∈ T, the set is well-ordered by the relation <. In particular, each well-ordered set is a tree. For each t ∈ T, the order type of is called the height of t. The height of T itself is the least ordinalgreater than the height of each element ofT. A root of a tree T is an element of height 0. Frequently trees are assumed to have only one root. Note that trees in set theory are often defined to grow downward making the root the greatest node. Trees with a single root may be viewed as rooted trees in the sense of graph theory in one of two ways: either as a tree or as a trivially perfect graph. In the first case, the graph is the undirected Hasse Diagram of the partially ordered set, and in the second case, the graph is simply the underlying graph of the partially ordered set. However, if T is a tree of height > ω, then the Hasse diagram definition does not work. For example, the partially ordered set does not have a Hasse Diagram, as there is no predecessor to ω. Hence we require height at most omega in this case. A branch of a tree is a maximal chain in the tree. The length of a branch is the ordinal that is order isomorphic to the branch. For each ordinal α, the α-th level of T is the set of all elements of T of height α. A tree is a κ-tree, for an ordinal number κ, if and only if it has height κ and every level has size less than the cardinality of κ. The width of a tree is the supremum of the cardinalities of its levels. Any single-rooted tree of height forms a meet-semilattice, where meet is given by maximal element of intersection of ancestors, which exists as the set of ancestors is non-empty and finite well-ordered, hence has a maximal element. Without a single root, the intersection of parents can be empty, for example where the elements are not comparable; while if there are an infinite number of ancestors there need not be a maximal element – for example, where are not comparable. A subtree of a tree is a tree where and is downward closed under, i.e., if and then.
Set-theoretic properties
There are some fairly simply stated yet hard problems in infinite tree theory. Examples of this are the Kurepa conjecture and the Suslin conjecture. Both of these problems are known to be independent of Zermelo–Fraenkel set theory. Kőnig's lemma states that every ω-tree has an infinite branch. On the other hand, it is a theorem of ZFC that there are uncountable trees with no uncountable branches and no uncountable levels; such trees are known as Aronszajn trees. A κ-Suslin tree is a tree of height κ which has no chains or antichains of size κ. In particular, if κ is singular then there exists a κ-Aronszajn tree and a κ-Suslin tree. In fact, for any infinite cardinal κ, every κ-Suslin tree is a κ-Aronszajn tree. The Suslin conjecture was originally stated as a question about certain total orderings but it is equivalent to the statement: Every tree of height ω1 has an antichain of cardinality ω1 or a branch of length ω1.
