The state-of-the-art graph wavelet decomposition was constructed by maximum spanning tree (MST)-based downsampling and two-channel graph wavelet filter banks. In this work, we first show that: 1) the existing MST-based downsampling could become unbalanced, i.e., the sampling rate is far from 1/2, which eventually leads to low representation efficiency of the wavelet decomposition; and 2) not only low-pass components, but also some high-pass ones can be decomposed to potentially achieve better decomposition performance.