Idea: use binary trees to implement the Set interface, such that doing a search for an element is like doing binary search.
interface Set<T> {
void insert(T e);
void remove();
boolean member(T e); // aka contains
...
}
The Binary-Search-Tree Property: For every node x in the tree,
y.data < x.data
(or y.data <= x.data
for MultiSet, that is, allow duplicates), andx.data < z.data
.We can also use BSTs to implement the Map interface (aka. “dictionary”).
interface Map<K,V> {
V get(K key);
V put(K key, V value);
V remove(K key);
boolean containsKey(K key);
}
BinarySearchTree
is like BinaryTree
(has a root
) but also
has a lessThan
predicate for comparing elements.
We define Node
as a class nested inside BinarySearchTree
for
convenience:
public class BinarySearchTree<K> implements Set<K> {
static class Node<K> {
K data;
Node<K> left, right;
// ...
}
Node<K> root;
protected BiPredicate<K, K> lessThan;
BinarySearchTree(BiPredicate<K, K> less) { lessThan = less; }
// ...
}
find
, search
, and contains
methods of BinarySearchTree
Book 4.3.1.
Example: Search for 6, 9, 15 in the following tree:
8
/ \
/ \
3 10
/ \ \
1 6 14
/ \ /
4 7 13
The find()
method looks for the node with the specified key; if there is none,
it returns the parent of where such a node would be.
protected Node<K> find(K key, Node<K> curr, Node<K> parent) {
if (curr == null)
return parent;
else if (lessThan.test(key, curr.data))
return find(key, curr.left, curr);
else if (lessThan.test(curr.data, key))
return find(key, curr.right, curr);
else
return curr;
}
The search()
method looks for the specified key and returns the node
if the key is found and otherwise returns null
.
public Node<K> search(K key) {
Node<K> n = find(key, root, null);
if (n == null)
return null;
else if (n.data.equals(key)))
return n;
else
return null;
}
The contains()
method returns true if the key is found and false otherwise.
public boolean contains(K key) {
Node<K> p = search(key);
return p != null;
}
What is the time complexity? O(h), where h is the height of the tree.
insert
method of BinarySearchTree
Textbook 4.3.3.
Similarly we can perform insertions on BSTs. The insert()
method
adds the given key to the tree.
public void insert_rec(K key) {
root = insert_helper(key, root);
}
protected static Node<K> insert_helper(K key, Node<K> curr) {
if (curr == null)
return new Node<>(key, null, null);
else if (lessThan.test(key, curr.data)) {
curr.left = insert_helper(key, curr.left);
return curr;
} else if (lessThan.test(curr.data, key)) {
curr.right = insert_helper(key, curr.right);
return curr;
} else {
// duplicate; do nothing
return curr;
}
}
What is the time complexity? O(h), where h is the height of the tree.
Insert into a binary search tree, returning the inserted node, or null if the key is already in the tree.
This can be accomplished using the find
method to do most of the
work of finding the location for the insert.
Fill in the blanks:
public Node<K> insert(K key) {
Node<K> n = find(key, root, null);
if (n == null){
} else if (lessThan.test(key, n.data)) {
} else if (lessThan.test(n.data, key)) {
} else {
}
}
remove
method of BinarySearchTree
Book 4.3.4.
| |
z=o A
\ ==>
A
| |
z=o A
/ ==>
A
|
z=o
/ \
A B
The main idea is to replace z with the node after z, which is the first node y in subtree B. We then recursively delete y from B.
What is the time complexity? Let h be the height of the tree. The time complexity is O(h).
Solution for remove()
(similar to book Figure 4.25):
public void remove(K key) {
root = remove_helper(root, key);
}
private Node remove_helper(Node<K> curr, K key) {
if (curr == null) {
return null;
} else if (lessThan.test(key, curr.data)) { // remove in left subtree
curr.left = remove_helper(curr.left, key);
return curr;
} else if (lessThan.test(curr.data, key)) { // remove in right subtree
curr.right = remove_helper(curr.right, key);
return curr;
} else { // remove this node
if (curr.left == null) {
return curr.right;
} else if (curr.right == null) {
return curr.left;
} else { // two children, replace with first of right subtree
Node<K> min = curr.right.first();
curr.data = min.data;
curr.right = remove_helper(curr.right, min.data);
return curr;
}
}
}