The Suffix Tree, a crucial and versatile data structure for string analysis of large texts, is often used in pattern matching and in bioinformatics applications. The Affix Tree generalizes the Suffix Tree in that it supports full tree functionalities in both search directions. The bottleneck of Affix Trees is their space requirement for storing the data structure. Here, we discuss existing representations and classify them into two categories: Synchronous and Asynchronous. We design Compressed Affix Tree indexes in both categories and explored how to support all tree operations bidirectionally. This work compares alternative approaches for compress the Affix Tree, measuring their space and time trade-offs for different operations. Moreover, to our knowledge, this is the first work that compares all Compressed Affix Tree implementations, i.e., four different approaches: the Affix Array, the Bidirectional Wavelet Index, and our two new structures, measures their space and time trade-offs, offering a practical benchmark for this structure.