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

10.1 Binary Search Trees
All of the structures we discuss in this chapter are search trees, that is, tree data
structures that can be used to implement a dictionary. Let us, therefore, begin by
briefly reviewing the fundamental methods of the dictionary ADT:
• find(k): Return an entry with key k, if it exists.
• findAll(k): Return an iterable collection of all entries with keys equal to k.
• insert(k,x): Insert an entry with key k and value x.
• remove(e): Remove an entry e, and return it.
• removeAll(k): Remove all entries with key k, returning an iterator of their
values.
Method find returns null if k is not found. The ordered dictionary ADT includes
some additional methods for searching through predecessors and successors of a key
or entry, but their performance is similar to that of find. So we will be focusing on
find as the primary search operation in this chapter.
Binary trees are an excellent data structure for storing the entries of a dictionary,
assuming we have an order relation defined on the keys. As mentioned previously
(Section 7.3.6), a binary search tree is a binary tree T such that each internal node v
of T stores an entry (k,x) such that:
• Keys stored at nodes in the left subtree of v are less than or equal to k.
• Keys stored at nodes in the right subtree of v are greater than or equal to k.
As we show below, the keys stored at the nodes of T provide a way of performing a
search by making a comparison at each internal node v, which can stop at v or
continue at v's left or right child. Thus, we take the view here that binary search trees
are nonempty proper binary trees. That is, we store entries only at the internal nodes
of a binary search tree, and the external nodes serve only as "placeholders." This
approach simplifies several of our search and update algorithms. Incidentally, we
could have allowed for improper binary search trees, which have better space usage,
but at the expense of more complicated search and update methods.
Independent of whether we view binary search trees as proper or not, the important
property of a binary search tree is the realization of an ordered dictionary (or map).
That is, a binary search tree should hierarchically represent an ordering of its keys,
using relationships between parent and children. Specifically, an inorder traversal
(Section 7.3.6) of the nodes of a binary search tree T should visit the keys in
nondecreasing order.
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