Phylogenetic combinatorics deals with the combinatorial aspects of 
phylogenetic-tree reconstruction. A starting point was the 
observation that, given a finite collection X of taxa, there is a 
close relationship between phylogenetic X-trees, certain metrics
D defined on X, and weighted compatible systems of X-splits.
In my lecture, I will focus on some rather new developments within
this context relating to block decomposition and virtual cut points 
of metric spaces that allow to canonically decompose any given 
finite metric space into a sum of pairwise compatible block metrics 
thus providing a far-reaching generalization of the result referred 
to above.