We conduct a comprehensive study of the effects of incorporating galaxy morphology information in photometric redshift estimation. Using machine learning methods, we assess the changes in the scatter and catastrophic outlier fraction of photometric redshifts when galaxy size, ellipticity, S{e}rsic index and surface brightness are included in training on galaxy samples from the SDSS and the CFHT Stripe-82 Survey (CS82). We show that by adding galaxy morphological parameters to full $ugriz$ photometry, only mild improvements are obtained, while the gains are substantial in cases where fewer passbands are available. For instance, the combination of $grz$ photometry and morphological parameters almost fully recovers the metrics of $5$-band photometric redshifts. We demonstrate that with morphology it is possible to determine useful redshift distribution $N(z)$ of galaxy samples without any colour information. We also find that the inclusion of quasar redshifts and associated object sizes in training improves the quality of photometric redshift catalogues, compensating for the lack of a good star-galaxy separator. We further show that morphological information can mitigate biases and scatter due to bad photometry. As an application, we derive both point estimates and posterior distributions of redshifts for the official CS82 catalogue, training on morphology and SDSS Stripe-82 $ugriz$ bands when available. Our redshifts yield a 68th percentile error of $0.058(1+z)$, and a catastrophic outlier fraction of $5.2$ per cent. We further include a deep extension trained on morphology and single $i$-band CS82 photometry.
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