Compressive sensing (CS) can be applied to fast wavelet-encoded MRI in order to reconstruct images from sparse sampling at sub-Nyquist sampling rates. In this paper, sparse MRI is achieved by a well-designed sampling matrix that satisfies the restricted isometry property (RIP) in CS. The wavelet-tree structure in k-space is utilized in order to reduce the RIP constant. The undersampling of k-space is implemented in a MRI simulator by spatially-selective RF excitation pulses, which are designed as Battle-Lemarie wavelet functions. The resulting undersampled k-space contains many significant wavelet-encoded samples and an improved RIP. The experimental results show that the proposed CS-MRI scheme reduced the number of necessary measurements significantly with the same reconstruction precision achieved by the current Fourier-encoded CS-MRI.