Editing Treap (section)

==Randomized binary search tree==
The randomized binary search tree, introduced by Martínez and Roura subsequently to the work of Aragon and Seidel on treaps,<ref>{{Citation | title=Randomized binary search trees | journal=Journal of the ACM | volume=45 | issue=2 | year=1997 | first1=Conrado | last1=Martínez | first2=Salvador | last2=Roura | pages=288–323 | url=http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.243 | doi=10.1145/274787.274812| s2cid=714621 | doi-access=free }}</ref> stores the same nodes with the same random distribution of tree shape, but maintains different information within the nodes of the tree in order to maintain its randomized structure.

Rather than storing random priorities on each node, the randomized binary search tree stores a small integer at each node, the number of its descendants (counting itself as one); these numbers may be maintained during tree rotation operations at only a constant additional amount of time per rotation. When a key ''x'' is to be inserted into a tree that already has ''n'' nodes, the insertion algorithm chooses with probability 1/(''n''&nbsp;+&nbsp;1) to place ''x'' as the new root of the tree, and otherwise, it calls the insertion procedure recursively to insert ''x'' within the left or right subtree (depending on whether its key is less than or greater than the root). The numbers of descendants are used by the algorithm to calculate the necessary probabilities for the random choices at each step. Placing ''x'' at the root of a subtree may be performed either as in the treap by inserting it at a leaf and then rotating it upwards, or by an alternative algorithm described by Martínez and Roura that splits the subtree into two pieces to be used as the left and right children of the new node.

The deletion procedure for a randomized binary search tree uses the same information per node as the insertion procedure, but unlike the insertion procedure, it only needs on average O(1) random decisions to join the two subtrees descending from the left and right children of the deleted node into a single tree. That is because the subtrees to be joined are on average at depth Θ(log n); joining two trees of size n and m needs Θ(log(n+m)) random choices on average. If the left or right subtree of the node to be deleted is empty, the join operation is trivial; otherwise, the left or right child of the deleted node is selected as the new subtree root with probability proportional to its number of descendants, and the join proceeds recursively.

===Comparison===
The information stored per node in the randomized binary tree is simpler than in a treap (a small integer rather than a high-precision random number), but it makes a greater number of calls to the random number generator (O(log&nbsp;''n'') calls per insertion or deletion rather than one call per insertion) and the insertion procedure is slightly more complicated due to the need to update the numbers of descendants per node. A minor technical difference is that, in a treap, there is a small probability of a collision (two keys getting the same priority), and in both cases, there will be statistical differences between a true random number generator and the [[Pseudorandom number generator|pseudo-random number generator]] typically used on digital computers. However, in any case, the differences between the theoretical model of perfect random choices used to design the algorithm and the capabilities of actual random number generators are vanishingly small.

Although the treap and the randomized binary search tree both have the same random distribution of tree shapes after each update, the history of modifications to the trees performed by these two data structures over a sequence of insertion and deletion operations may be different. For instance, in a treap, if the three numbers 1, 2, and 3 are inserted in the order 1, 3, 2, and then the number 2 is deleted, the remaining two nodes will have the same parent-child relationship that they did prior to the insertion of the middle number. In a randomized binary search tree, the tree after the deletion is equally likely to be either of the two possible trees on its two nodes, independently of what the tree looked like prior to the insertion of the middle number.