Hypersuccinct Trees -- New universal tree source codes for optimal compressed tree data structures.

We present a new universal source code for unlabeled binary and ordinal trees that achieves asymptotically optimal compression for all tree sources covered by existing universal codes. At the same time, it supports answering many navigational queries on the compressed representation in constant time on the word-RAM; this is not known to be possible for any existing tree compression method. The resulting data structures, "hypersuccinct trees", hence combine the compression achieved by the best known universal codes with the operation support of the best succinct tree data structures. Compared to prior work on succinct data structures, we do not have to tailor our data structure to specific applications; hypersuccinct trees automatically adapt to the trees at hand. We show that it simultaneously achieves the asymptotically optimal space usage for a wide range of distributions over tree shapes, including: random binary search trees (BSTs) / Cartesian trees of random arrays, random fringe-balanced BSTs, binary trees with a given number of binary/unary/leaf nodes, random binary tries generated from memoryless sources, full binary trees, unary paths, as well as uniformly chosen weight-balanced BSTs, AVL trees, and left-leaning red-black trees. Using hypersuccinct trees, we further obtain the first data structure that answers range-minimum queries on a random permutation of $n$ elements in constant time and using the optimal $1.736n + o(n)$ bits on average, solving an open problem of Davoodi et al. (2014) and Golin et al. (2016).