  We explore phases of free fermions on arenas that do not tessellate a
manifold. Specializing to arboreal arenas described by tree graphs which
possess a notion of translation symmetry, we study possible fermionic phases in
the BDI symmetry class on the $p$-coordinated Bethe lattice. We find that there
are $p$ distinct obstructed atomic insulating phases that are characterized by
distinct edge states, pattern of entanglement, and a winding characteristic
that we define here. These distinct insulting phases are always separated by a
metallic region in the parameter space rather than isolated quantum critical
points. The metallic region itself comprises several distinct metallic phases
that are distinguished by the winding characteristic and correlation functions.
The correlation functions of distinct subsystems display non-analytic behavior
at distinct points in the metallic region, signaling a cascade of subsystem
transitions. An intriguing feature of these arboreal metals is the presence of
truncated subsystems with zero energy boundary modes despite being gapless.
This work suggests new opportunities for synthetic quantum systems to realize
these novel phases.
