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tokens: of their stellar mass at the present was already in place by z ∼ 2 via star formation (see Fig. 8 ). Figure 9. The average stellar mass assembly history of central galaxies as a function of redshift for haloes of different present-day masses, predicted by various empirical models as indicated. For Milky Way sized haloes, however, the differences between the model predictions are milder. All the models predict that about half of stellar mass was in place by z ∼ 1. There is a significant difference between L15 -U and other models at high z . For example, L15 -U predicts that about 15 per cent of the final stellar mass was assembled by z ∼ 2, while less than 10 per cent was predicted by the other models. For central galaxies in present-day massive haloes with M h (0) > 10 14 M ⊙ , the predictions of different models again become very different. M13 predicts a much later assembly for these galaxies than any other models. The predictions of B13 look similar to L15 -U, but the increase in stellar mass with time is due to different reasons. While L15 -U predicts that the increase at z 2 is dominated by accretion of stars from satellites, B13 predicts that a significant fraction of the increase at z 2 is actually due to in situ star formation (see Fig. 8 ). This difference is again due to the boost of star formation in low-mass haloes at high z in the L15 -U model. The increased amount of stars formed in progenitors at high z makes the accretion of stars more important in the growth of stellar mass in a massive galaxy, and the fraction of stars formed in situ has to be decreased proportionally in order to match the final stellar mass of the galaxy. The results demonstrate the importance of properly modelling the star formation in low-mass progenitors at high z in order to understand the star formation and stellar mass assembly histories of massive galaxies at the present day. 4.4 Stellar mass functions of high-redshift galaxies Fig. 10 shows the predictions of the empirical models for the SMFs of galaxies at a number of redshifts. The predictions of B13 and M13 are similar in both slopes and amplitudes at the low-mass ends, but B13 predicts many more massive galaxies than M13 , particularly at high redshifts. Y12 predicts significantly flatter slopes at the low-mass ends, and more galaxies in the intermediate mass range, than the other three models. The SMFs predicted by the L15 -U model match the predictions of B13 at M * > 10 10 M ⊙ , but are significantly steeper at the low-mass ends. Figure 10. The model predictions for the field SMFs at high redshifts (solid) in comparison with observations. We select some observational SMFs of high-redshift galaxies from the literature to compare with the model predictions. Specifically, we use the SMFs at 1.3 z 3.5 given by Pérez-González et al. ( 2008 ) and Marchesini et al. ( 2009 ). Pérez-González et al. used a sample combining data in three different fields with a total area of 664 arcmin 2 that have a total