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The range-minimum query (RMQ) problem is a fundamental data structuring taskwith numerous applications. Despite the fact that succinct solutions withworst-case optimal $2n+o(n)$ bits of space and constant query time are known,it has been unknown whether such a data structure can be made adaptive to thereduced entropy of random inputs (Davoodi et al. 2014). We construct a succinctdata structure with the optimal $1.736n+o(n)$ bits of space on average forrandom RMQ instances, settling this open problem.  Our solution relies on a compressed data structure for binary trees that isof independent interest. It can store a (static) binary search tree generatedby random insertions in asymptotically optimal expected space and supports manyqueries in constant time. Using an instance-optimal encoding of subtrees, wefurthermore obtain a "hyper-succinct" data structure for binary trees thatimproves upon the ultra-succinct representation of Jansson, Sadakane and Sung(2012).

Entropy Trees and Range-Minimum Queries In Optimal Average-Case Space