since the last access of $x$. Whereas dynamic finger captures proximity in keyspace, the working set property captures proximity *in time*. In 2001, Iacono [@IaconoUnified] proposed a *unified* property that generalizes both kinds of proximity. Informally, a data structure with the unified property is efficient when accessing an item that is *close to some recently accessed item*. [It is not known whether any BST data structure has the unified property.]{}

Recently, Iacono and Langerman [@LI16] studied the *lazy finger* property (Bose et al. [@BoseDIL14]), and showed that an online algorithm called Greedy BST[^3] satisfies it. The lazy finger property requires the amortized cost of accessing $x$ to be $O(d)$, where $d$ is the distance (number of edges) from the previously accessed item to $x$ in the best *static reference tree*. This property is stronger than the dynamic finger property [@BoseDIL14], and it is not known to hold for Splay.

In this paper we study a generalization of the lazy finger property; instead of a single finger stationed at the previously accessed item, we allow $k$ fingers to be moved around arbitrarily. An access is performed by moving any of the fingers to the requested item. Cost is proportional to the 