Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

10.2 AVL Trees
In the previous section, we discussed what should be an efficient dictionary data
structure, but the worst-case performance it achieves for the various operations is
linear time, which is no better than the performance of list- and array-based dictionary
implementations (such as unordered lists and search tables discussed in Chapter 9). In
this section, we describe a simple way of correcting this problem so as to achieve
logarithmic time for all the fundamental dictionary operations.
Definition of an AVL Tree
The simple correction is to add a rule to the binary search tree definition that will
maintain a logarithmic height for the tree. The rule we consider in this section is the
following height-balance property, which characterizes the structure of a binary
search tree T in terms of the heights of its internal nodes (recall from Section 7.2.1
that the height of a node v in a tree is the length of a longest path from v to an
external node):
Height-Balance Property: For every internal node v of T, the heights of the
children
of v differ by at most 1.
Any binary search tree T that satisfies the height-balance property is said to be an
AVL tree, named after the initials of its inventors: Adel'son-Vel'skii and Landis. An
example of an AVL tree is shown in Figure 10.7.
Figure 10.7: An example of an AVL tree. The keys of
the entries are shown inside the nodes, and the heights
of the nodes are shown next to the nodes.
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