Binary search trees are designed for efficient access to data. They are built to be search engines that locate
an element along a path from the root. In an application, the actual efficiency depends on the shape of the
tree. The order in which data enters the tree may cause a subtree to be heavily weighted to one side or the
other. In a worst case, the tree is degenerate or "almost degenerate" where most of the n elements are
stored as a lone child of a parent. The shape resembles a linked list (Figure 1(a)) and has search efficiency
O(n). The other extreme is a complete binary tree that stores the n elements in a tree of minimum height by
uniformly distributing the nodes in the left and right subtrees. Access to any element requires no more than
int(log2 n) + 1 comparisons and the search efficiency is O(log2 n). A complete tree represents an ideal shape
for a search tree.
A normal binary search tree uses search tree ordering to insert an element. The add() method
follows a rigid set of rules that locates an element as a leaf node without regard for the overall shape of the
tree. We need search trees that use a dynamic insert algorithm that rearranges elements whenever a subtree
get out of balance. The goal is to have a search tree with a measure of balance among the subtrees similar
to a complete tree. Over the years, researchers have developed just such search trees. In this document, we
will discuss AVL search trees and red-black search trees. An AVL tree, named after its discoverers
Adelson, Velskii and Landis, uses recursive insert and delete algorithms that maintain height-balance at
each node. By this we mean that for each node, the difference in height of its two subtrees is in the range -1
to 1. Figure 1(b) is an AVL tree. For node 70, the height of its left subtree is 1 and the height of its right
subtree is 2. The difference heightL - heightR = -1 and the tree is "slightly tilted" to the right. In contrast to a
simple binary search tree, an AVL tree can never become heavily weight to one side or the other. A red-
black search tree provides a different kind a structure and balance criteria. The design of a red-black tree
has its origins in a balanced search tree called a 2-3-4 tree. The description comes from the fact that each
node has 2, 3, or 4 links (children). A 2-3-4 tree is perfectly balanced in the sense that no interior node has
a null child and all leaf nodes are at the same level in the tree (Figure 1(d)). A red-black tree provides a
representation of 2-3-4 trees. The trees feature nodes that have the color attribute BLACK or RED. The tree
maintains a measure of balance called the BLACK-height. Figure 1(c) is a representation of the 2-3-4 tree
with RED nodes displayed with shading. It is BLACK-height balanced since the path from the root to any
empty subtree has two black node. This concept will become clear when we introduce red-black trees in
Section 4. Modern data structures use red-black trees to implement ordered sets and map such as the
TreeSet and TreeMap collection classes. In Section 7, we create the RBTree class that implements the red-
black tree balancing algorithms.
FIGURE 1
Different search tree structures for the list {50, 95, 60, 90, 70, 80, 75, 78}.
Binary search
trees are
designed for
efficient location
of an element.
AVL and
red-black trees
balance a binary
search tree so it
more nearly
resembles a
complete tree.
Binary Se arch Tree (a)
50
78
75
80
70
90
60
80
75
7850
9060
70
AVL Tree (b)
60, 75, 90
50 70
2-3-4 Tree (d)
78, 80 95
Red-Black Tree (c)
78
807050
9060
75
95
95 95
