The utilization of wavelet-based techniques in graph neural networks (GNNs) has gained considerable attention, particularly in the context of node classification. Although existing wavelet-based approaches have shown promise, they are constrained by their reliance on pre-defined wavelet filters, rendering them incapable of effectively adapting to signals that reside on graphs based on tasks at hand. Recent research endeavors address this issue through the introduction of a wavelet lifting transform. However, this technique necessitates the use of bipartite graphs, causing a transformation of the original graph structure into a bipartite configuration. This alteration of graph topology results in the generation of undesirable wavelet filters, thereby undermining the effectiveness of the method. In response to these challenges, we propose a novel simple and effective adaptive graph wavelet neural network (SEA-GWNN) class that employs the lifting scheme on arbitrary graph structures while upholding the original graph topology by leveraging multi-hop computation trees. A noteworthy aspect of the approach is the focus on local substructures represented as acyclic trees, wherein the lifting strategy is applied in a localized manner. This locally defined lifting scheme effectively combines high-pass and low-pass frequency information to enhance node representations. Furthermore, to reduce computing costs, we propose to decouple the higher-order lifting operators and induce them from the lower-order structures. Finally, we benchmark our model on several real-world datasets spanning four distinct categories, including citation networks, webpages, the film industry, and large-scale graphs and the experimental results showcase the efficacy of the proposed SEA-GWNN.