  In their work on `Coxeter-like complexes', Babson and Reiner introduced a
simplicial complex $\Delta_T$ associated to each tree $T$ on $n$ nodes,
generalizing chessboard complexes and type A Coxeter complexes. They
conjectured that $\Delta_T$ is $(n-b-1)$-connected when the tree has $b$
leaves. We provide a shelling for the $(n-b)$-skeleton of $\Delta_T$, thereby
proving this conjecture.
  In the process, we introduce notions of weak order and inversion functions on
the labellings of a tree $T$ which imply shellability of $\Delta_T$, and we
construct such inversion functions for a large enough class of trees to deduce
the aforementioned conjecture and also recover the shellability of chessboard
complexes $M_{m,n}$ with $n \ge 2m-1$. We also prove that the existence or
nonexistence of an inversion function for a fixed tree governs which networks
with a tree structure admit greedy sorting algorithms by inversion elimination
and provide an inversion function for trees where each vertex has capacity at
least its degree minus one.
