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We define simple variants of zip trees, called zip-zip trees, which provideseveral advantages over zip trees, including overcoming a bias that favorssmaller keys over larger ones. We analyze zip-zip trees theoretically andempirically, showing, e.g., that the expected depth of a node in an $n$-nodezip-zip tree is at most $1.3863\log n-1+o(1)$, which matches the expected depthof treaps and binary search trees built by uniformly random insertions. Unlikethese other data structures, however, zip-zip trees achieve their bounds usingonly $O(\log\log n)$ bits of metadata per node, w.h.p., as compared to the$\Theta(\log n)$ bits per node required by treaps. In fact, we even describe a``just-in-time'' zip-zip tree variant, which needs just an expected $O(1)$number of bits of metadata per node. Moreover, we can define zip-zip trees tobe strongly history independent, whereas treaps are generally only weaklyhistory independent. We also introduce \emph{biased zip-zip trees}, which havean explicit bias based on key weights, so the expected depth of a key, $k$,with weight, $w_k$, is $O(\log (W/w_k))$, where $W$ is the weight of all keysin the weighted zip-zip tree. Finally, we show that one can easily make zip-ziptrees partially persistent with only $O(n)$ space overhead w.h.p.

Zip-zip Trees: Making Zip Trees More Balanced, Biased, Compact, or Persistent