  In this paper we study functions on the interval that have the same
persistent homology. By introducing an equivalence relation modeled after
topological conjugacy, which we call graph-equivalence, a precise enumeration
of functions with the same persistent homology is given, inviting comparisons
with Arnold's Calculus of Snakes. The equivalence classes used here are indexed
by chiral merge trees, which are binary merge trees where a left-right ordering
of the children of each vertex is given. Enumeration of merge trees and chiral
merge trees with the same persistence makes essential use of the Elder Rule (a
criterion for pairing critical points), which is given a new proof here as
well.
