Entropy and Optimal Compression of Some General Plane Trees

We continue developing the information theory of structured data. In this paper, we take up d-ary trees (d ≥ 2) and trees with unrestricted degree. We rst compute the entropy which gives us the fundamental lower bound on compression of such trees. Then we present e cient compression algorithms based on arithmetic encoding that achieve the entropy within a constant number of bits. A naïve implementation of these algorithms has a prohibitive time complexity of O(nd ) elementary operations, but our e cient algorithms run in O(n2) of these operations, where n is the number of nodes. It turns out that extending source coding (i.e., compression) from sequences to advanced data structures such as general trees is mathematically quite challenging and leads to new recurrences that nd ample applications in the information theory of general structures (e.g., to analyze the information content of general non-plane trees).