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With the new operations, the implementation of AVL trees can be more efficient and highly-parallelizable. [13] The function Join on two AVL trees t 1 and t 2 and a key k will return a tree containing all elements in t 1, t 2 as well as k. It requires k to be greater than all keys in t 1 and smaller than all keys in t 2.
Join follows the right spine of t 1 until a node c which is balanced with t 2. At this point a new node with left child c, root k and right child t 2 is created to replace c. The new node may invalidate the balancing invariant. This can be fixed with rotations. The following is the join algorithms on different balancing schemes. The join ...
CGAL : Computational Geometry Algorithms Library in C++ contains a robust implementation of Range Trees; Boost.Icl offers C++ implementations of interval sets and maps. IntervalTree (Python) - a centered interval tree with AVL balancing, compatible with tagged intervals; Interval Tree (C#) - an augmented interval tree, with AVL balancing
AA tree; AVL tree; Binary search tree; Binary tree; Cartesian tree; Conc-tree list; Left-child right-sibling binary tree; Order statistic tree; Pagoda; Randomized binary search tree; Red–black tree; Rope; Scapegoat tree; Self-balancing binary search tree; Splay tree; T-tree; Tango tree; Threaded binary tree; Top tree; Treap; WAVL tree; Weight ...
To turn a regular search tree into an order statistic tree, the nodes of the tree need to store one additional value, which is the size of the subtree rooted at that node (i.e., the number of nodes below it). All operations that modify the tree must adjust this information to preserve the invariant that size[x] = size[left[x]] + size[right[x]] + 1
For comparison, an AVL tree is guaranteed to be within a factor of 1.44 of the optimal height while requiring only two additional bits of storage in a naive implementation. [1] Therefore, most self-balancing BST algorithms keep the height within a constant factor of this lower bound.
However, this introduces extra complexity into the implementation and may cause even worse performance for smaller hash tables, where the time spent inserting into and balancing the tree is greater than the time needed to perform a linear search on all elements of a linked list or similar data structure.
This makes scapegoat trees easier to implement and, due to data structure alignment, can reduce node overhead by up to one-third. Instead of the small incremental rebalancing operations used by most balanced tree algorithms, scapegoat trees rarely but expensively choose a "scapegoat" and completely rebuilds the subtree rooted at the scapegoat ...