We have studied the phase separation of a binary mixture of a polymer and a nematic liquid crystal at a dilute polymer concentration. Various types of regular self-assemblies of polymer droplets such as branched chains and zigzag chains have been observed in addition to the previously reported straight chains. The kinetic process of the self-assembly is dominated by the transformations of a topological defect accompanied by a droplet in addition to the simple growth of the droplet. The spontaneous transformation from a ring defect (Saturn-ring defect) to a point defect (dipole) is an essential process in forming a straight chain composed of droplets with the same size. We also find that the transformations are induced by a nearby droplet with a point defect. This leads to a wide size distribution of droplets in a chain cluster and results in a branched chain. The difference in the chaining mechanisms is discussed using an electrostatic analogy. We also clarify that the symmetry breaking in the assembly is governed by the direction in which the cell containing the binary mixture is rubbed.