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 .