Because reactivity increases as particle size decreases and competition between numerous structures are possible, which affects catalytic and magnetic properties, we study the structural trends of late-transition-metal 13-atom clusters using density functional theory within the generalized gradient approximation to exchange-correlation functional. We consider open structural motifs, such as bilayer and cubic (recently found to have lower energy), and find new bilayer candidates that are even lower in energy. To study the influence of d-orbital filling on structural trends, we focus on Pt, Pd, and Rh clusters and find several new, low-energy structures for Pt13 and Pd13 from searches using a first-principle molecular dynamics high-temperature annealing. We find that 13-atom clusters prefer less square-cubic order as the d-orbitals gradually fill in the order of Rh, Pt, and Pd, and generally, low symmetry, open structures are preferred to high symmetry, compact ones, a trend explained from their electronic structures. For completeness, we briefly comment on improved exchange-correlation functionals affects on cluster morphology.