  Inspired by the interval decomposition of persistence modules and the
extended Newick format of phylogenetic networks, we show that, inside the
larger category of \textit{ordered Reeb graphs}, every Reeb graph with $n$
leaves and first Betti number $s$, is equal to a coproduct of at most $2^s$
trees with $(n + s)$ leaves. Reeb graphs are therefore classified up to
isomorphism by their tree decomposition. An implication of this result, is that
the isomorphism problem for Reeb graphs is fixed parameter tractable when the
parameter is the first Betti number. We propose ordered Reeb graphs as a model
for time consistent phylogenetic networks and propose a certain Hausdorff
distance as a metric on these structures.
