In the entropy coding, AIVF (almost instantaneous variable-to-fixed length) codes using multiple parsing trees can attain a better compression rate than the Tunstall code, which attains the best compression rate in the class of VF codes with a single parsing tree. Furthermore, the multiple parsing trees of an AIVF code can be multiplexed into a single parsing tree. In this paper, we propose a new universal data compression code based on the techniques of the AIVF code. The proposed universal code can also be considered as an improvement of the LZW code (Welch code). We explain how the AIVF coding techniques can be applied to universal coding by growing dynamically a single parsing tree, and we evaluate the compression rate of the proposed universal code theoretically and using several corpora.