  We revisit the classical problem of searching in a binary search tree (BST)
using rotations, and present novel connections of this problem to a number of
geometric and combinatorial structures. In particular, we show that the
execution trace of a BST that serves a sequence of queries is in close
correspondence with the flip-sequence between two rectangulations.
(Rectangulations are well-studied combinatorial objects also known as mosaic
floorplans.) We also reinterpret Small Manhattan Network, a problem with known
connections to the BST problem, in terms of flips in rectangulations. We apply
further transformations to the obtained geometric model, to arrive at a
particularly simple view of the BST problem that resembles sequences of
edge-relaxations in a shortest path algorithm.
  Our connections yield new results and observations for all structures
concerned. In this draft we present some preliminary findings. BSTs with
rotations are among the most fundamental and most thoroughly studied objects in
computer science, nonetheless they pose long-standing open questions, such as
the dynamic optimality conjecture of Sleator and Tarjan (STOC 1983). Our hope
is that the correspondences presented in this paper provide a new perspective
on this old problem and bring new tools to the study of dynamic optimality.
