${\mbox{\sf SO}}(X) = \min_R \sum_{i} d_R(x_i)$. Here $R$ is a *static* BST, and $d_R(x)$ is the depth of $x$ in $R$.

For the *dynamic finger* property [@ST85], ${\mbox{\sf DF}}(X) =\sum_t \log |x_t - x_{t+1}|$, and for *working set* [@ST85], ${\mbox{\sf WS}}(X) = \sum_t \log \rho_t(x_t)$, where $\rho_t(a)$ is the number of distinct keys accessed between time $t$ and the last time at which $a$ was accessed (all keys assumed accessed at time zero).

In 2001, Iacono [@IaconoUnified] initiated the study of a property that would “unify” the latter two notions of efficiency and exhibited a data structure (not a BST) achieving this property. This *unified bound* is defined as ${\mbox{\sf UB}}(X) = \sum_{t} \min_{t' < t} \log (|x_t - x_{t'}| + \rho_{t}(x_{t'}))$. Dynamic finger and working set are in general, not comparable. On the other hand, ${\mbox{\sf UB}}(X) \leq {\mbox{\sf DF}}(X)$, and ${\mbox{\sf UB}}(X) \leq {\mbox{\sf WS}}(X)$ clearly hold, justifying the name of the unified bound.

Despite several attempts, the question whether the unified bound is a valid BST property remains unclear; it was shown in [@DerryberryS09] that ${\mbox{\sf OPT}}(X) = O({\mbox{\sf UB}}(X) +m\log \log n)$, and in [@unified; @IaconoUnified] that the unified bound is valid in some other (non-BST) 