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

Method
Time
size,isEmpty
O(1)
find, insert, remove
O(h)
O(h + s)
findAll
Note that the running time of search and update operations in a binary search tree
varies dramatically depending on the tree's height. We can nevertheless take
comfort that, on average, a binary search tree with n keys generated from a
random series of insertions and removals of keys has expected height O(logn).
Such a statement requires careful mathematical language to precisely define what
we mean by a random series of insertions and removals, and sophisticated
probability theory to prove; hence, its justification is beyond the scope of this
book. Nevertheless, keep in mind the poor worst-case performance and take care
in using standard binary search trees in applications where updates are not
random. There are, after all, applications where it is essential to have a dictionary
with fast worst-case search and update times. The data structures presented in the
next sections address this need.
10.1.3 Java Implementation
In Code Fragments 10.3 through 10.5, we describe a binary search tree class,
BinarySearchTree, which stores objects of class BSTEntry (implementing
the Entry interface) at its nodes. Class BinarySearchTree extends class
Linked BinaryTree from Code Fragments 7.16 through 7.18, thus taking
advantage of code reuse.
This class makes use of several auxiliary methods to do much of the heavy lifting.
The auxiliary method treeSearch, based on the TreeSearch algorithm (Code
Fragment 10.1), is invoked by the find, findAll, and insert methods. We
use a recursive addAll method as the main engine for the findAll(k) method, in
that it performs an inorder traversal of all the entries with keys equal to k (although
not using the fast algorithm, since it performs a failed search for every entry it
finds). We use two additional update methods, insertAtExternal, which
inserts a new entry at an external node, and removeExternal, which removes an
external node and its parent.
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