Using first-principles density-functional calculations, we investigate the structure and growth mechanism of allyl mercaptan lines on the H-terminated Si(100)-2$\ifmmode\times\else\texttimes\fi{}$1 surface. The earlier structural model (termed the linear structure), where the terminal C atom of the C=C bond initially reacts with a single Si dangling bond, has been competing with a new structural model (termed the branched structure) where the medial C atom of the C=C bond initially reacts with a single Si dangling bond. We find that formation of the branched structure is kinetically unfavored over that of the linear structure. Moreover, the simulated scanning tunneling microscopy (STM) image of the branched structure cannot reproduce the features observed in the STM experiment, such as the position and origin of the bright protrusion. Thus, the present results do not support the branched structure in many respects.