On $C^0$-persistent homology and trees

The study of the topology of the superlevel sets of stochastic processes on $[0,1]$ in probability led to the introduction of $\R$-trees which characterize the connected components of the superlevel-sets. We provide a generalization of this construction to more general deterministic continuous functions on some path-connected, compact topological set $X$ and reconcile the probabilistic approach with the traditional methods of persistent homology. We provide an algorithm which functorially links the tree $T_f$ associated to a function $f$ and study some invariants of these trees, which in 1D turn out to be linked to the regularity of $f$.