The range-minimum query (RMQ) problem is a fundamental data structuring task with numerous applications. Despite the fact that succinct solutions with worst-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 the reduced entropy of random inputs (Davoodi et al. 2014). We construct a succinct data structure with the optimal 1.736n + o(n) bits of space on average for random RMQ instances, settling this open problem.Our solution relies on a compressed data structure for binary trees that is of independent interest. It can store a (static) binary search tree generated by random insertions in asymptotically optimal expected space and supports many queries in constant time. Using an instance-optimal encoding of subtrees, we furthermore obtain a "hyper-succinct" data structure for binary trees that improves upon the ultra-succinct representation of Jansson, Sadakane and Sung (2012).1 To simplify the presentation, we assume the elements in A are unique. In the general case, we fix a tie-breaking rule, usually to return the leftmost minimum. Our data structures extend to any such convention.

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