The radial evolution of the power spectra of the MHD turbulence within the trailing edge of high-speed streams in the solar wind has been investigated with the magnetic field data of HELIOS 1 and 2 for heliocentric distances between 0.3 and 0.9 AU. In the analyzed frequency range 2.8 ⋅ 10-4 8.3 ⋅ 10-2 Hz the computed spectra have, near the earth, values of the spectral index close to that predicted for an imcompressible hydromagnetic turbulence in a stationary state. Approaching the sun, the spectral slope remains unchanged for frequencies f?10-2 Hz, whereas at lower frequencies we find a clear evolution toward a less steep falloff with frequency. The radial gradient of the power in Alfv¿nic fluctuations depends on frequency and it increases upon increasing frequency. For frequencies f?10-2 Hz, however, the radial gradient remains approximately the same. A discussion of possible theoretical implications of the observational features pointed out is given.