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Concept
Random binary tree
Summary
In computer science and probability theory, a random binary tree is a binary tree selected at random from some probability distribution on binary trees. Two different distributions are commonly used: binary trees formed by inserting nodes one at a time according to a random permutation, and binary trees chosen from a uniform discrete distribution in which all distinct trees are equally likely. It is also possible to form other distributions, for instance by repeated splitting. Adding and removing nodes directly in a random binary tree will in general disrupt its random structure, but the treap and related randomized binary search tree data structures use the principle of binary trees formed from a random permutation in order to maintain a balanced binary search tree dynamically as nodes are inserted and deleted.
For random trees that are not necessarily binary, see random tree.
For any set of numbers (or, more generally, values from some total order), one may form a binary search tree in which each number is inserted in sequence as a leaf of the tree, without changing the structure of the previously inserted numbers. The position into which each number should be inserted is uniquely determined by a binary search in the tree formed by the previous numbers. For instance, if the three numbers (1,3,2) are inserted into a tree in that sequence, the number 1 will sit at the root of the tree, the number 3 will be placed as its right child, and the number 2 as the left child of the number 3. There are six different permutations of the numbers (1,2,3), but only five trees may be constructed from them. That is because the permutations (2,1,3) and (2,3,1) form the same tree.
For any fixed choice of a value x in a given set of n numbers, if one randomly permutes the numbers and forms a binary tree from them as described above, the expected value of the length of the path from the root of the tree to x is at most 2 log n + O(1), where "log" denotes the natural logarithm function and the O introduces big O notation.
Official source
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