  In this paper we face the problem of representation of functional data with
the tools of algebraic topology. We represent functions by means of merge trees
and this representation is compared with that offered by persistence diagrams.
We show that these two structures, although not equivalent, are both invariant
under homeomorphic re-parametrizations of the functions they represent, thus
allowing for a statistical analysis which is indifferent to functional
misalignment. We employ a novel metric for merge trees and we prove some
theoretical results related to its specific implementation when merge trees
represent functions. To showcase the good properties of our topological
approach to functional data analysis, we test it on the Aneurisk65 dataset
replicating, from our different perspective, the supervised classification
analysis which contributed to make this dataset a benchmark for methods dealing
with misaligned functional data.
