  This article studies the limit of binary search trees drawn from Mallows
permutations under various topologies. The main result, pertaining to the
standard local topology for graphs, requires the introduction of a
generalization of binary search trees to two-sided infinite sequences, referred
to as \textit{redwood trees}. We then show that the almost-sure local limit of
finite Mallows trees is the redwood tree drawn from the two-sided infinite
Mallows permutation, thus corresponding to swapping the local limit and the
binary search tree structure.
  Building off this result, we study various other natural topologies: the
rooted topology of the local structure around the root, the
Gromov-Hausdorff-Prokhorov topology of the tree seen as a metric space, and the
subtree size topology of the ratio of nodes split between left and right
subtrees. The limit of Mallows trees under these three topologies combined with
the case of the local topology allow us to draw a global picture of what large
Mallows trees look like from different perspective and further strengthen the
relation between finite and infinite Mallows permutations.
