Context. Leo A is a gas-rich dwarf irregular galaxy of low stellar mass located in the outskirts of the Local Group. It has an extended star formation history with stellar populations spanning a wide age range (∼0.01−10 Gyr). As Leo A is a well-isolated dwarf galaxy, it is a perfect target to study a galactic structure formed entirely by processes of self-induced star formation. Aims. Our aim is to study populations of the brightest asymptotic giant branch (AGB) stars and red giant branch (RGB) stars over the entire extent of the Leo A galaxy. Methods. We analysed populations of AGB and RGB stars in the Leo A galaxy using multicolour photometry data obtained with the Subaru Suprime-Cam (B, V, R, I, Hα) and HST ACS (F475W, F814W) cameras. In order to separate the Milky Way and Leo A populations of red stars, we developed a photometric method that enabled us to study the spatial distribution of AGB and RGB stars within the Leo A galaxy. Results. We found a previously unknown sequence of 26 peculiar RGB stars which probably have a strong CN band in their spectra (∼380−390 nm). This conclusion is supported by the infrared CN spectral features observed in four of these stars with available spectra from the literature. Additionally, we present a catalogue of 32 luminous AGB stars and 3 candidate AGB stars. Twelve AGB stars (three of them might have dusty envelopes) from this sample are newly identified; the remaining 20 AGB stars were already presented in the literature based on near-infrared observations. By splitting the RGB sequence into blue and red parts, we revealed different spatial distributions of the two subsets, with the former being more centrally concentrated than the latter. Cross-identification with spectroscopic data available in the literature suggests that the bulk of blue and red RGB stars are, on average, similar in metallicity; however, the red RGB stars might have an excess of metal-deficient stars of [Fe/H] < −1.8. We also found that the distributions of luminous AGB and blue RGB stars have nearly equal scale lengths (0.′87 ± 0.′06 and 0.′89 ± 0.′09, respectively), indicating that they could belong to the same generation. This conclusion is strengthened by the similarities of the cumulative distributions of AGB and blue RGB stars, both showing more centrally concentrated populations compared to red RGB stars. There is also a prominent decline in the ratio of AGB to RGB stars with an increasing radius. These results suggest that the star-forming disk of Leo A is shrinking, which is in agreement with the outside-in star formation scenario of dwarf galaxy evolution.