We propose a new metric for rooted, labelled, binary phylogenies which distinguishes trees in a different way from existing phylogenetic metrics. In particular, our metric captures something of the difference in the `biological story’ told by the trees, with an emphasis on comparing shape, root placement, lineages and, if desired, branch lengths. We will outline some of the applications where our metric has already proven useful. These include the comparison of tree inference methods, comparing gene trees to full sequence trees, sorting the trees from an MCMC chain, and visualising tree space, which enables us to detect `islands’ (collections of trees with similar likelihood and topology). Finally we will describe some work in progress where we use the metric to find informative sites in sequence alignments. This is joint work with Caroline Colijn.