Randomly encoded compressive sensing (CS) has potential in fast acquisition of magnetic resonance imaging (MRI) data in most naturally compressible images. However, there is no guaranteed good performance for general applications by any of the traditional CS-MRI theoretical schemes developed so far. On the other hand, recent research demonstrates that adaptive sampling exploiting the tree structure of nonzero wavelet coefficients of signals allows more control over the sensing procedure in the form of feedback and improve the CS performance significantly. Following recent research strategies in CS-MRI, well-known adaptive sampling strategies in the wavelet domain, as used in image compression, to encode the MRI data yielding good reconstruction quality are introduced. Based on this underlying characteristic, adaptive k-space trajectories are designed with tailored spatially selective RF excitation pulses created by Battle-Lemarie wavelet functions. The input vectors formed from these significant samples of multilevel wavelet decomposed images are used in a CS framework for reconstruction of MR images. This MR image reconstruction uses a CS algorithm based on the minimization of total-variation regularized signal to provide stable results. The simulated results show that this approach can reduce almost 70% of MR image acquisition time and achieve good reconstructed image quality.