Abstract
A tree-child network is a phylogenetic network with the property that each non-leaf vertex is the parent of a tree vertex or a leaf. In this paper, we show that a tree-child network on taxa (leaf) set X with an outgroup and a positive real-valued weighting of its edges is essentially determined by the multi-set of all path-length distances between elements in X provided, for each reticulation, the edges directed into it have equal weight. Furthermore, we give a polynomial-time algorithm for reconstructing such a network from this inter-taxa distance information. Such constructions are of central importance in evolutionary biology where phylogenetic networks represent the ancestral history of a collection of present-day taxa.
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