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A labeled binary tree of size 9 (the number of nodes in the tree) and height 3 (the height of a tree defined as the number of edges or links from the top-most or root node to the farthest leaf node), with a root node whose value is 1. The above tree is unbalanced and not sorted.
The size of an internal node is the sum of sizes of its two children, plus one: (size[n] = size[n.left] + size[n.right] + 1). Based on the size, one defines the weight to be weight[n] = size[n] + 1. [a] Weight has the advantage that the weight of a node is simply the sum of the weights of its left and right children. Binary tree rotations.
The B-tree generalizes the binary search tree, allowing for nodes with more than two children. [2] Unlike other self-balancing binary search trees, the B-tree is well suited for storage systems that read and write relatively large blocks of data, such as databases and file systems.
Since k-d trees divide the range of a domain in half at each level of the tree, they are useful for performing range searches. Analyses of binary search trees has found that the worst case time for range search in a k-dimensional k-d tree containing n nodes is given by the following equation. [13]
The search operation, in a standard trie, takes () time, where is the size of the string parameter , and corresponds to the alphabet size. [16]: 754 Binary search trees, on the other hand, take () in the worst case, since the search depends on the height of the tree () of the BST (in case of balanced trees), where and being number of keys ...
This unsorted tree has non-unique values (e.g., the value 2 existing in different nodes, not in a single node only) and is non-binary (only up to two children nodes per parent node in a binary tree). The root node at the top (with the value 2 here), has no parent as it is the highest in the tree hierarchy.
Most operations on a binary search tree (BST) take time directly proportional to the height of the tree, so it is desirable to keep the height small. A binary tree with height h can contain at most 2 0 +2 1 +···+2 h = 2 h+1 −1 nodes. It follows that for any tree with n nodes and height h: + And that implies:
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