In this paper, we show an effective way of using adaptive self-organizing data structures in enhancing compression schemes. We introduce a new data structure, the Partitioning Binary Search Tree (PBST), which is based on the well-known Binary Search Tree (BST), and when used in conjunction with Fano encoding, the PBST leads to the so-called Fano Binary Search Tree (FBST). The PBST and FBST can be maintained adaptively and in a self-organizing manner by using new tree-based operators, namely the Shift-To-Left (STL) and the Shift-To-Right (STR) operators. The encoding and decoding procedures that also update the FBST have been implemented, and show that the adaptive Fano coding using FBSTs, the Huffman, and the greedy adaptive Fano coding achieve similar compression ratios.