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This idea was applied by {{harvtxt|Seidel|Aragon|1996}}, who suggested the use of random numbers as priorities. The [[self-balancing binary search tree]] resulting from this random choice is called a [[treap]], due to its combination of binary search tree and min-heap features. An insertion into a treap can be performed by inserting the new key as a leaf of an existing tree, choosing a priority for it, and then performing [[tree rotation]] operations along a path from the node to the root of the tree to repair any violations of the heap property caused by this insertion; a deletion can similarly be performed by a constant amount of change to the tree followed by a sequence of rotations along a single path in the tree.{{sfnp|Seidel|Aragon|1996}} A variation on this data structure called a zip tree uses the same idea of random priorities, but simplifies the random generation of the priorities, and performs insertions and deletions in a different way, by splitting the sequence and its associated Cartesian tree into two subsequences and two trees and then recombining them.{{sfnp|Tarjan|Levy|Timmel|2021}}
If the priorities of each key are chosen randomly and independently once whenever the key is inserted into the tree, the resulting Cartesian tree will have the same properties as a [[random binary search tree]], a tree computed by inserting the keys in a randomly chosen [[permutation]] starting from an empty tree, with each insertion leaving the previous tree structure unchanged and inserting the new node as a leaf of the tree. Random binary search trees had been studied for much longer, and are known to behave well as search trees. In particular,
===In sorting===
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