We present a variety of structures formed by colloidal droplets at a nematic liquid crystal - air interface, where the elastic dipole-dipole, quadrupole-quadrupole and dipole-quadrupole interactions are all essentially involved. The colloidal structures observed not only include chains with kinks or clusters but also comprise directional structures, such as directional chains and branches, whose direction is associated with the tilting director in the liquid crystal layer. The dipole-quadrupole interaction, originating from the polydispersity of the droplets, plays a central role for the formation of these directional structures. Clusters consisting of directional branches and chains are also observed and found to be fractal statistically.