  Square-root topology is a recently emerged subfield describing a class of
insulators and superconductors whose topological nature is only revealed upon
squaring their Hamiltonians, i.e., the finite energy edge states of the
starting square-root model inherit their topological features from the
zero-energy edge states of a known topological insulator/superconductor present
in the squared model. Focusing on one-dimensional models, we show how this
concept can be generalized to $2^n$-root topological insulators and
superconductors, with $n$ any positive integer, whose rules of construction are
systematized here. Borrowing from graph theory, we introduce the concept of
arborescence of $2^n$-root topological insulators/superconductors which
connects the Hamiltonian of the starting model for any $n$, through a series of
squaring operations followed by constant energy shifts, to the Hamiltonian of
the known topological insulator/superconductor, identified as the source of its
topological features. Our work paves the way for an extension of $2^n$-root
topology to higher-dimensional systems.
