Modeling studies indicate that double-headed streamers originating from single electron avalanches in lightning-driven quasi-static electric fields at mesospheric altitudes accelerate and expand, reaching transverse scales from tens to a few hundreds of meters and propagation speeds up to one tenth of the speed of light, in good agreement with recent telescopic, high-speed video and multichannel photometric observations of sprites. The preionization of the medium ahead of a streamer by the ionizing UV photons originating from a region of high electric field in the streamer head (i.e., photoionization) significantly modifies the streamer scaling properties as a function of air pressure in comparison with those predicted by similarity laws. The photoionization leads to lower peak electric fields in the streamer head, lower streamer electron densities, wider initial streamer structures, and lower acceleration and expansion rates of streamers at sprite altitudes 40--90 km, when compared to the ground level. The primary reason for the observed differences is that the effective quenching altitude of the excited states of the molecular nitrogen b1Πu, b'1$Sigma_{u}^{+}$, and c'41$Sigma_{u}^{+}$ that give photoionizing radiation is about 24 km. The quenching of these states is therefore negligible at sprite altitudes, leading to a substantial enhancement of the electron-ion pair production ahead of the streamer tip because of the photoionization, when compared to the ground level. The maximum radius of the expanding streamers is predominantly controlled by the combination of the absorption cross section χmin = 3.5 ¿ 10-2 cm-1 Torr-1 of the molecular oxygen (O2) at 1025 ¿ and the partial pressure of O2 in air, pO2. Streamers exhibit branching when their radius becomes greater than 1/χminpO2. Model results indicate a lower branching threshold radius for positive streamers in comparison with negative streamers, under otherwise identical ambient conditions. These results are in good agreement with recent results of high-speed photography of laboratory streamers in near-atmospheric pressure N2/O2 mixtures and similar morphology documented during recent telescopic and high-speed video observations of sprites.