Metrics on multi-labelled trees: interrelationships and diameter bounds

Multi-labeled trees or MUL-trees, for short, are trees whose leaves are labeled by elements of some non-empty finite set $X$ such that more than one leaf may be labeled by the same element of $X$. This class of trees includes phylogenetic trees and tree shapes. MUL-trees arise naturally in, for example, biogeography and gene evolution studies and also in the area of phylogenetic network reconstruction. In this paper we introduce novel metrics which may be used to compare MUL-trees, most of which generalize well-known metrics on phylogenetic trees and tree shapes. These metrics can be used, for example, to better understand the space of MUL-trees or to help visualize collections of MUL-trees. In addition, we describe some relationships between the MUL-tree metrics that we present and also give some novel diameter bounds for these metrics. We conclude by briefly discussing some open problems, as well as pointing out how MUL-tree metrics may be used to define metrics on the space of phylogenetic networks.