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 $\beta$. In this paper we define a related coefficient $\alpha$ describing the cost of splay-to-root traversal on maximal (i.e., complete) binary trees, and show that $\beta \geq 2 + \alpha$. We give the first 3009 digits of $\alpha$, including the decimal point, and show that every digit is correct. We make two conjectures: first, that $\beta = 2 + \alpha$, and second, that $\alpha$ is irrational.