Tree space nerds (I'm looking at you, @mathmomike and @david_bryant) ---
This morning I stumbled across this paper, which uses discrete Morse theory to analyze a space of "tree with symmetries". This paper has gotten almost no citations in the phylogenetics literature, but I think it shows some methods that may be interesting for folks interested in the large scale structure of tree space:
The topology of spaces of phylogenetic trees with symmetry ☆ Axel
Natural Dowling analogues of the complex of phylogenetic trees are
studied. Using discrete Morse theory, we find their homotopy types. In
the process, the homotopy types of certain subposets of Dowling
lattices are determined.