a sequence of keys [is]{} accessed in a binary search tree (BST), and after each access, the tree can be reconfigured via a sequence of rotations and pointer moves starting from the root. (There exist several alternative but essentially equivalent models, see [@Wilber; @DHIKP09].) Denote the space of keys (or elements) by $[n]$. For a sequence $X = (x_1,\ldots, x_m) \in [n]^m$, denote by ${\sf OPT}(X)$ the cost of the optimal offline BST for accessing $X$.[^5] Arguably the most important question in the BST model is the dynamic optimality conjecture, i.e. the existence of an online BST whose cost is $O({\mbox{\sf OPT}}(X))$ for every $X$.

A BST [*optimality property*]{} is an inequality between ${\sf OPT}(X)$ and some function $f(X)$, that holds in the BST model. (If ${\sf OPT}(X) \leq f(X)$ for all $X$ is a BST optimality property, then every $O(1)$-competitive algorithm must cost at most $O(f(X))$.)

Several natural BST properties have been suggested over the last few decades. For instance, the *static finger* property [@ST85] states ${\mbox{\sf OPT}}(X) =O({\mbox{\sf SF}}(X))$, for ${\mbox{\sf SF}}(X) = \sum_t \log |x_t - j|$, where $j \in [n]$ is a fixed element (finger). The *static optimality* property [@ST85] is ${\mbox{\sf OPT}}(X) =O({\mbox{\sf SO}}(X))$, where 