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, 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 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, but only five trees may be constructed from them. That is because the permutations and form the same tree.
Expected depth of a node
For any fixed choice of a value in a given set of 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 is at most, where "" denotes the natural logarithm function and the introduces big O notation. For, the expected number of ancestors of is by linearity of expectation equal to the sum, over all other values in the set, of the probability that is an ancestor of. And a value is an ancestor of exactly when is the first element to be inserted from the elements in the interval. Thus, the values that are adjacent to in the sorted sequence of values have probability of being an ancestor of, the values one step away have probability, etc. Adding these probabilities for all positions in the sorted sequence gives twice a Harmonic number, leading to the bound above. A bound of this form holds also for the expected search length of a path to a fixed value that is not part of the given set.
Although not as easy to analyze as the average path length, there has also been much research on determining the expectation of the length of the longest path in a binary search tree generated from a random insertion order. It is now known that this length, for a tree with nodes, is almost surely where is the unique number in the range satisfying the equation
In the random permutation model, each of the numbers from the set of numbers used to form the tree, except for the smallest and largest of the numbers, has probability of being a leaf in the tree, for it is a leaf when it inserted after its two neighbors, and any of the six permutations of these two neighbors and it are equally likely. By similar reasoning, the smallest and largest of the numbers have probability of being a leaf. Therefore, the expected number of leaves is the sum of these probabilities, which for is exactly.
In applications of binary search tree data structures, it is rare for the values in the tree to be inserted without deletion in a random order, limiting the direct applications of random binary trees. However, algorithm designers have devised data structures that allow insertions and deletions to be performed in a binary search tree, at each step maintaining as an invariant the property that the shape of the tree is a random variable with the same distribution as a random binary search tree. If a given set of ordered numbers is assigned numeric priorities, these priorities may be used to construct a Cartesian tree for the numbers, a binary tree that has as its inorder traversal sequence the sorted sequence of the numbers and that is heap-ordered by priorities. Although more efficient construction algorithms are known, it is helpful to think of a Cartesian tree as being constructed by inserting the given numbers into a binary search tree in priority order. Thus, by choosing the priorities either to be a set of independent random real numbers in the unit interval, or by choosing them to be a random permutation of the numbers from to , and by maintaining the heap ordering property using tree rotations after any insertion or deletion of a node, it is possible to maintain a data structure that behaves like a random binary search tree. Such a data structure is known as a treap or a randomized binary search tree.
Uniformly random binary trees
The number of binary trees with n nodes is a Catalan number: for these numbers of trees are Thus, if one of these trees is selected uniformly at random, its probability is the reciprocal of a Catalan number. Trees in this model have expected depth proportional to the square root of, rather than to the logarithm; however, the Strahler number of a uniformly random binary tree, a more sensitive measure of the distance from a leaf in which a node has Strahler number whenever it has either a child with that number or two children with number, is with high probability logarithmic. Due to their large heights, this model of equiprobable random trees is not generally used for binary search trees, but it has been applied to problems of modeling the parse trees of algebraic expressions in compiler design and for modeling evolutionary trees. In some cases the analysis of random binary trees under the random permutation model can be automatically transferred to the uniform model.
Random split trees
generate random binary trees with nodes by generating a real-valued random variable in the unit interval, assigning the first nodes to the left subtree, the next node to the root, and the remaining nodes to the right subtree, and continuing recursively in each subtree. If is chosen uniformly at random in the interval, the result is the same as the random binary search tree generated by a random permutation of the nodes, as any node is equally likely to be chosen as root; however, this formulation allows other distributions to be used instead. For instance, in the uniformly random binary tree model, once a root is fixed each of its two subtrees must also be uniformly random, so the uniformly random model may also be generated by a different choice of distribution for. As Devroye and Kruszewski show, by choosing a beta distribution on and by using an appropriate choice of shape to draw each of the branches, the mathematical trees generated by this process can be used to create realistic-looking botanical trees.
