The free energy landscape of the structures and dissociation degree of the first water and heavy water adlayers on Ru(0001) surface is presented. Thermodynamically favored interconversion routes connecting different experimentally reported structures are suggested based on free energy calculations. On going from low to high water coverage, one-dimensional (1D) periodic chain-like structures with small, or zero, water dissociation degree like Chain-4a motifs are found to be very stable intermediates in the formation of dissociated and molecular ice-like bilayers, respectively. The isotopic effects on the dissociation degree of the ice-like bilayers are estimated: a preference for the half- dissociated form is found for H2O at temperatures below 275 K followed by energetic degeneration between all bilayers dissociated over that threshold. Instead, for heavy water this temperature is shifted to 225 K. Moreover, the configurational entropy due to the different arrangements of dissociated and molecular flat molecules further contributes to set the energies of all these structures within a small energy window that make experimental identification difficult.