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English: Analysis of data structures, tree compared to hash and array based structures, height balanced tree compared to more perfectly balanced trees, a simple height balanced tree class with test code, comparable statistics for tree performance, statistics of worst case strictly-AVL-balanced trees versus perfect full binary trees.
Height-Balanced. BATON is considered balanced if and only if the height of its two sub-trees at any node in the tree differs by at most one. If any node detects that the height-balanced constraint is violated, a restructuring process is initiated to ensure that the tree remains balanced.
A weight-balanced tree is a binary search tree that stores the sizes of subtrees in the nodes. That is, a node has fields key, of any ordered type; value (optional, only for mappings) left, right, pointer to node; size, of type integer. By definition, the size of a leaf (typically represented by a nil pointer) is zero.
It depends on the balance factor of the sibling Z (the higher child tree in figure 2) whether the height of the subtree decreases by one –and the retracing needs to continue– or does not change (if Z has the balance factor 0) and the whole tree is in AVL-shape.
For height-balanced binary trees, the height is defined to be logarithmic () in the number of items. This is the case for many binary search trees, such as AVL trees and red–black trees . Splay trees and treaps are self-balancing but not height-balanced, as their height is not guaranteed to be logarithmic in the number of items.
Let h ≥ –1 be the height of the classic B-tree (see Tree (data structure) § Terminology for the tree height definition). Let n ≥ 0 be the number of entries in the tree. Let m be the maximum number of children a node can have. Each node can have at most m−1 keys.
The height of an external node is zero, and the height of any internal node is always one plus the maximum of the heights of its two children. Thus, the height function of an AVL tree obeys the constraints of a WAVL tree, and we may convert any AVL tree into a WAVL tree by using the height of each node as its rank. [1] [2]
If the two trees are balanced, join simply creates a new node with left subtree t 1, root k and right subtree t 2. Suppose that t 1 is heavier (this "heavier" depends on the balancing scheme) than t 2 (the other case is symmetric). Join follows the right spine of t 1 until a node c which is balanced with t 2.