A partially ordered set P has the fixed point property if every order-preserving map f : P → P has a fixed point, i.e. there exists x ∊ P such that f(x) = x. A. Tarski's classical result (see [4]), that every complete lattice has the fixed point property, is based on the following two properties of a complete lattice P:(A)For every order-preserving map f : P → P there exists x ∊ P such that x ≦ f(x).(B)Suprema of subsets of P exist; in particular, the supremum of the set {x|x ≦ f(x)} ⊂ P exists.

In this paper we prove that if the nodes of an arbitrary n-node binary search tree T are splayed according to the preorder sequence of T then the total time is O(n). This is a special case of the splay tree traversal conjecture of Sleator and Tarjan [1]. IntroductionA binary search tree in which we splay after each access to the node containing the accessed item is called a splay tree. Splaying is a restructuring operation consisting of a sequence of rotations (see [1] for details). Tarjan [2] proved that the nodes of an n-node search tree can be splayed in symmetric order (inorder) in O(n) time.

ABSTRACT. In this paper we prove that if the nodes of an arbitrary n-node binary search tree T are splayed in the preorder sequence of T then the total time is O(n). This is a special case of the splay tree traversal conjecture of Sleator and Tarjan.

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