Evolutionary multitask algorithms that adopt multitask optimization paradigms have been proposed to tackle multiple problems simultaneously and improve the performance of traditional evolutionary algorithms. One of the most crucial challenges in evolutionary multitasking applied to network design is the lack of an efficient unified representation to encode solutions. This paper presents the first representation based on node-depth encoding for evolutionary multitask algorithms to tackle network design problems. Remarkably, we propose an encoding method to represent solutions modeled by trees of arbitrary graphs in the form of a unified representation and design a corresponding decoding method to reconstruct solutions from a unified search space for each task. To verify the efficiency of our proposed methods, extensive experiments are conducted on well-known network design problems and demonstrate that our approach performs significantly better than previous approaches regarding solution quality.