  The canonical join complex of a semidistributive lattice is a simplicial
complex whose faces are canonical join representations of elements of the
semidistributive lattice. We give a combinatorial classification of the faces
of the canonical join complex of the lattice of biclosed sets of segments
supported by a tree, as introduced by the third author and McConville. We also
use our classification to describe the elements of the shard intersection order
of the lattice of biclosed sets. As a consequence, we prove that this shard
intersection order is a lattice.
