A coefficient related to splay-to-root traversal, correct to thousands of decimal places
This paper takes another look at the cost of traversing a binary tree using repeated splay-to-root. This was shown to cost $O(n)$ (in rotations) by Tarjan and later, in different ways, by Elmasry and others. It would be interesting to know the minimal possible coefficient implied by the $O(n)$ cost; call this coefficient $β$. In this paper we define a related coefficient $α$ describing the cost of splay-to-root traversal on maximal (i.e., complete) binary trees, and show that $β\geq 2 + α$. We give the first 3009 digits of $α$, including the decimal point, and show that every digit is correct. We make two conjectures: first, that $β= 2 + α$, and second, that $α$ is irrational.
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