In this thesis we study a specific combinatorial optimization problem, called the Balanced Minimum Evolution Problem (BMEP), which is an APX-hard network design problem that consists of finding a minimum length unrooted binary tree (called a phylogeny) having as a leaf-set a given set of molecular sequences. The optimal solution to the BMEP (i.e., the optimal phylogeny) encodes the hierarchical evolutionary relationships of the input sequences. This information is crucial for a multitude of research fields, ranging from systematics to medical research, passing through drug discovery, epidemiology, ecology, biodiversity assessment and population dynamics. We provide new results on the combinatorial and computational aspects of the BMEP, present an algorithm that performs better than previously known exact solution algorithms for the BMEP and discuss the implications of these insights for other phylogenetic estimation problems based on minimum evolution.