  Rooted phylogenetic networks provide an explicit representation of the
evolutionary history of a set $X$ of sampled species. In contrast to
phylogenetic trees which show only speciation events, networks can also
accommodate reticulate processes (for example, hybrid evolution, endosymbiosis,
and lateral gene transfer). A major goal in systematic biology is to infer
evolutionary relationships, and while phylogenetic trees can be uniquely
determined from various simple combinatorial data on $X$, for networks the
reconstruction question is much more subtle. Here we ask when can a network be
uniquely reconstructed from its `ancestral profile' (the number of paths from
each ancestral vertex to each element in $X$). We show that reconstruction
holds (even within the class of all networks) for a class of networks we call
`orchard networks', and we provide a polynomial-time algorithm for
reconstructing any orchard network from its ancestral profile. Our approach
relies on establishing a structural theorem for orchard networks, which also
provides for a fast (polynomial-time) algorithm to test if any given network is
of orchard type. Since the class of orchard networks includes tree-sibling
tree-consistent networks and tree-child networks, our result generalise
reconstruction results from 2008 and 2009. Orchard networks allow for an
unbounded number $k$ of reticulation vertices, in contrast to tree-sibling
tree-consistent networks and tree-child networks for which $k$ is at most
$2|X|-4$ and $|X|-1$, respectively.

Combinatorics