Circular orders of tree metrics, and their uses for the reconstruction and fitting of phylogenetic trees

The circular orders associated with planar drawings of an X-tree (or phylogenetic tree) have been studied by several authors, especially for the encoding of an X-tree by 2n-3 lengths of paths between leaves (n being the cardinality of X). It is shown here that circular orders are the same as those obtained from the table of a tree metric by a construction due to Yushmanov (1984). More, this construction applies to any dissimilarity, tree metric or not. Several fast algorithms (of complexity O(n2)) are derived from these results: for the determination of a Yushmanov order; for the reconstruction of the valued X-tree representing a tree metric; for the recognition of a tree metric; and for the adjustment of a tree metric to a given metric; the last one proceeds by successive local least squares approximations. It leads to a fitting procedure which, tested on some examples, gives interesting results.