Median genomes serve as candidates for ancestral reconstruction and help study the diversity of parsimonious rearrangement solutions. However, the breakpoint median problem is NP-hard for unichromosomal genomes, and medians are non-unique, making their diversity important to study. In this paper, we develop tree-based algorithms for studying breakpoint medians of linear unichromosomal genomes represented as unsigned permutations. The algorithms construct rooted labeled trees whose leaves encode candidate median genomes satisfying local adjacency constraints derived from the inputs. We show that the resulting reduced search spaces contain all breakpoint medians while being much smaller than the full permutation space. We investigate these reduced spaces experimentally for three-genome instances. Our method finds exact medians and many near-optimal approximations, while early sampling of leaves often produces near-medians quickly. We also introduce constrained sampling procedures for targeting balanced near-medians, whose distances to the input genomes are more evenly distributed and hence are particularly relevant for ancestral reconstruction. The resulting framework provides a unified method for exact enumeration, efficient sampling, and targeted exploration of median and near-median genomes. In contrast to Traveling Salesman Problem-based approaches, which typically return a single solution and do not reveal the diversity or balance of the median set, our method can enumerate all medians, sample near-medians early, and target structurally distinct candidates under the breakpoint distance.

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