性质：
使二叉树成为二叉查找树的性质是：对于树种每个节点X，它的左子树中的所有项的值小于X中的项，它的右子树中所有项的值大于X中的项
Api
BinarySearchTree-API.png
contains
public boolean contains(AnyType x){
return contains(x,root);
}
private boolean contains(AnyType x, BinaryNode<AnyType> t) {
if(t==null) return false;
int cmpResult = x.compareTo(t.element);
if(cmpResult<0){
return contains(x,t.left);
}else if(cmpResult > 0){
return contains(x,t.right);
}else{
return true;
}
}
findMin和findMax
public AnyType findMin(){
if(isEmpty())return null;
return findMin(root).element;
}
private BinaryNode<AnyType> findMin(BinaryNode<AnyType> t) {
if(t == null){
return null;
}else if(t.left == null){
return t;
}
return findMin(t.left);
}
public AnyType findMax(){
if(isEmpty())return null;
return findMax(root).element;
}
private BinaryNode<AnyType> findMax(BinaryNode<AnyType> t) {
if(t!=null){
while(t.right!=null){
t = t.right;
}
}
return t;
}
insert
public void insert(AnyType x){
root = insert(x,root);
}
private BinaryNode<AnyType> insert(AnyType x, BinaryNode<AnyType> t) {
if(t == null)
return new BinaryNode<AnyType>(x,null,null);
int cmpResult = x.compareTo(t.element);
if(cmpResult<0){
t.left = insert(x, t.left);
}else if(cmpResult>0){
t.right = insert(x, t.right);
}else{
//duplicate do nothing
}
return t;
}
remove
- 要删除的是叶子节点：直接删除
- 要删除的节点有一个子节点：调整父节点的链
- 要删除的节点有两个子节点：一般的策略是找到右子树的最小节点代替该节点并递归的删除那个最小节点，
为什么选择右子树的最小子节点：首先满足二叉查找树的性质，其次，第二次remove很容易，因为最小节点不可能有左子树，所以第二次删除就是一个简单的操作。
public void remove(AnyType x){
root = remove(x,root);
}
private BinaryNode<AnyType> remove(AnyType x, BinaryNode<AnyType> t) {
if(t == null)
return t;//item not found ,do nothing
int cmpResult = x.compareTo(t.element);
if(cmpResult<0){
t.left = remove(x,t.left);
}else if(cmpResult>0){
t.right = remove(x,t.right);
}else if(t.left !=null && t.right !=null ){
t.element = findMin(t.right).element;
t.right = remove(t.element,t.right);//
}else{
t = (t.left!=null)? t.left : t.right;
}
return t;
}
select
假设要找排名为k的节点（即树种有k个节点小于它的节点）
如果左子树中的节点数t大于k，就继续递归的在左子树中查找排名为k的节点
如果左子树中的节点数t等于k，就返回根节点
如果左子树中的节点数t大于k，就递归的在右子树中查找排名(k-t-1)的节点
public Key select(int k) {
if (k < 0 || k >= size()) throw new IllegalArgumentException();
Node x = select(root, k);
return x.key;
}
// Return key of rank k.
private Node select(Node x, int k) {
if (x == null) return null;
int t = size(x.left);
if (t > k) return select(x.left, k);
else if (t < k) return select(x.right, k-t-1);
else return x;
}
rank 返回给定节点的排名
如果给定键与根节点相等，就返回左子树的节点总数t
如果给定键小于根节点，就返回该键在左子树中的排名
如果给定键大于根节点，就返回t+1加上它在右子树中的排名
public int rank(Key key) {
if (key == null) throw new NullPointerException("argument to rank() is null");
return rank(key, root);
}
// Number of keys in the subtree less than key.
private int rank(Key key, Node x) {
if (x == null) return 0;
int cmp = key.compareTo(x.key);
if (cmp < 0) return rank(key, x.left);
else if (cmp > 0) return 1 + size(x.left) + rank(key, x.right);
else return size(x.left);
}
《算法第四版》的实现
public class BST<Key extends Comparable<Key>, Value> {
private Node root; // root of BST
private class Node {
private Key key; // sorted by key
private Value val; // associated data
private Node left, right; // left and right subtrees
private int size; // number of nodes in subtree
public Node(Key key, Value val, int size) {
this.key = key;
this.val = val;
this.size = size;
}
}
/**
* Initializes an empty symbol table.
*/
public BST() {
}
/**
* Returns true if this symbol table is empty.
* @return {@code true} if this symbol table is empty; {@code false} otherwise
*/
public boolean isEmpty() {
return size() == 0;
}
/**
* Returns the number of key-value pairs in this symbol table.
* @return the number of key-value pairs in this symbol table
*/
public int size() {
return size(root);
}
// return number of key-value pairs in BST rooted at x
private int size(Node x) {
if (x == null) return 0;
else return x.size;
}
/**
* Does this symbol table contain the given key?
*
* @param key the key
* @return {@code true} if this symbol table contains {@code key} and
* {@code false} otherwise
* @throws NullPointerException if {@code key} is {@code null}
*/
public boolean contains(Key key) {
if (key == null) throw new NullPointerException("argument to contains() is null");
return get(key) != null;
}
/**
* Returns the value associated with the given key.
*
* @param key the key
* @return the value associated with the given key if the key is in the symbol table
* and {@code null} if the key is not in the symbol table
* @throws NullPointerException if {@code key} is {@code null}
*/
public Value get(Key key) {
return get(root, key);
}
private Value get(Node x, Key key) {
if (x == null) return null;
int cmp = key.compareTo(x.key);
if (cmp < 0) return get(x.left, key);
else if (cmp > 0) return get(x.right, key);
else return x.val;
}
/**
* Inserts the specified key-value pair into the symbol table, overwriting the old
* value with the new value if the symbol table already contains the specified key.
* Deletes the specified key (and its associated value) from this symbol table
* if the specified value is {@code null}.
*
* @param key the key
* @param val the value
* @throws NullPointerException if {@code key} is {@code null}
*/
public void put(Key key, Value val) {
if (key == null) throw new NullPointerException("first argument to put() is null");
if (val == null) {
delete(key);
return;
}
root = put(root, key, val);
assert check();
}
private Node put(Node x, Key key, Value val) {
if (x == null) return new Node(key, val, 1);
int cmp = key.compareTo(x.key);
if (cmp < 0) x.left = put(x.left, key, val);
else if (cmp > 0) x.right = put(x.right, key, val);
else x.val = val;
x.size = 1 + size(x.left) + size(x.right);
return x;
}
/**
* Removes the smallest key and associated value from the symbol table.
*
* @throws NoSuchElementException if the symbol table is empty
*/
public void deleteMin() {
if (isEmpty()) throw new NoSuchElementException("Symbol table underflow");
root = deleteMin(root);
assert check();
}
private Node deleteMin(Node x) {
if (x.left == null) return x.right;
x.left = deleteMin(x.left);
x.size = size(x.left) + size(x.right) + 1;
return x;
}
/**
* Removes the largest key and associated value from the symbol table.
*
* @throws NoSuchElementException if the symbol table is empty
*/
public void deleteMax() {
if (isEmpty()) throw new NoSuchElementException("Symbol table underflow");
root = deleteMax(root);
assert check();
}
private Node deleteMax(Node x) {
if (x.right == null) return x.left;
x.right = deleteMax(x.right);
x.size = size(x.left) + size(x.right) + 1;
return x;
}
/**
* Removes the specified key and its associated value from this symbol table
* (if the key is in this symbol table).
*
* @param key the key
* @throws NullPointerException if {@code key} is {@code null}
*/
public void delete(Key key) {
if (key == null) throw new NullPointerException("argument to delete() is null");
root = delete(root, key);
assert check();
}
private Node delete(Node x, Key key) {
if (x == null) return null;
int cmp = key.compareTo(x.key);
if (cmp < 0) x.left = delete(x.left, key);
else if (cmp > 0) x.right = delete(x.right, key);
else {
if (x.right == null) return x.left;
if (x.left == null) return x.right;
Node t = x;
x = min(t.right);
x.right = deleteMin(t.right);
x.left = t.left;
}
x.size = size(x.left) + size(x.right) + 1;
return x;
}
/**
* Returns the smallest key in the symbol table.
*
* @return the smallest key in the symbol table
* @throws NoSuchElementException if the symbol table is empty
*/
public Key min() {
if (isEmpty()) throw new NoSuchElementException("called min() with empty symbol table");
return min(root).key;
}
private Node min(Node x) {
if (x.left == null) return x;
else return min(x.left);
}
/**
* Returns the largest key in the symbol table.
*
* @return the largest key in the symbol table
* @throws NoSuchElementException if the symbol table is empty
*/
public Key max() {
if (isEmpty()) throw new NoSuchElementException("called max() with empty symbol table");
return max(root).key;
}
private Node max(Node x) {
if (x.right == null) return x;
else return max(x.right);
}
/**
* Returns the largest key in the symbol table less than or equal to {@code key}.
*
* @param key the key
* @return the largest key in the symbol table less than or equal to {@code key}
* @throws NoSuchElementException if there is no such key
* @throws NullPointerException if {@code key} is {@code null}
*/
public Key floor(Key key) {
if (key == null) throw new NullPointerException("argument to floor() is null");
if (isEmpty()) throw new NoSuchElementException("called floor() with empty symbol table");
Node x = floor(root, key);
if (x == null) return null;
else return x.key;
}
private Node floor(Node x, Key key) {
if (x == null) return null;
int cmp = key.compareTo(x.key);
if (cmp == 0) return x;
if (cmp < 0) return floor(x.left, key);
Node t = floor(x.right, key);
if (t != null) return t;
else return x;
}
/**
* Returns the smallest key in the symbol table greater than or equal to {@code key}.
*
* @param key the key
* @return the smallest key in the symbol table greater than or equal to {@code key}
* @throws NoSuchElementException if there is no such key
* @throws NullPointerException if {@code key} is {@code null}
*/
public Key ceiling(Key key) {
if (key == null) throw new NullPointerException("argument to ceiling() is null");
if (isEmpty()) throw new NoSuchElementException("called ceiling() with empty symbol table");
Node x = ceiling(root, key);
if (x == null) return null;
else return x.key;
}
private Node ceiling(Node x, Key key) {
if (x == null) return null;
int cmp = key.compareTo(x.key);
if (cmp == 0) return x;
if (cmp < 0) {
Node t = ceiling(x.left, key);
if (t != null) return t;
else return x;
}
return ceiling(x.right, key);
}
/**
* Return the kth smallest key in the symbol table.
*
* @param k the order statistic
* @return the kth smallest key in the symbol table
* @throws IllegalArgumentException unless {@code k} is between 0 and
* <em>N</em> − 1
*/
public Key select(int k) {
if (k < 0 || k >= size()) throw new IllegalArgumentException();
Node x = select(root, k);
return x.key;
}
// Return key of rank k.
private Node select(Node x, int k) {
if (x == null) return null;
int t = size(x.left);
if (t > k) return select(x.left, k);
else if (t < k) return select(x.right, k-t-1);
else return x;
}
/**
* Return the number of keys in the symbol table strictly less than {@code key}.
*
* @param key the key
* @return the number of keys in the symbol table strictly less than {@code key}
* @throws NullPointerException if {@code key} is {@code null}
*/
public int rank(Key key) {
if (key == null) throw new NullPointerException("argument to rank() is null");
return rank(key, root);
}
// Number of keys in the subtree less than key.
private int rank(Key key, Node x) {
if (x == null) return 0;
int cmp = key.compareTo(x.key);
if (cmp < 0) return rank(key, x.left);
else if (cmp > 0) return 1 + size(x.left) + rank(key, x.right);
else return size(x.left);
}
/**
* Returns all keys in the symbol table as an {@code Iterable}.
* To iterate over all of the keys in the symbol table named {@code st},
* use the foreach notation: {@code for (Key key : st.keys())}.
*
* @return all keys in the symbol table
*/
public Iterable<Key> keys() {
return keys(min(), max());
}
/**
* Returns all keys in the symbol table in the given range,
* as an {@code Iterable}.
*
* @param lo minimum endpoint
* @param hi maximum endpoint
* @return all keys in the symbol table between {@code lo}
* (inclusive) and {@code hi} (inclusive)
* @throws NullPointerException if either {@code lo} or {@code hi}
* is {@code null}
*/
public Iterable<Key> keys(Key lo, Key hi) {
if (lo == null) throw new NullPointerException("first argument to keys() is null");
if (hi == null) throw new NullPointerException("second argument to keys() is null");
Queue<Key> queue = new Queue<Key>();
keys(root, queue, lo, hi);
return queue;
}
private void keys(Node x, Queue<Key> queue, Key lo, Key hi) {
if (x == null) return;
int cmplo = lo.compareTo(x.key);
int cmphi = hi.compareTo(x.key);
if (cmplo < 0) keys(x.left, queue, lo, hi);
if (cmplo <= 0 && cmphi >= 0) queue.enqueue(x.key);
if (cmphi > 0) keys(x.right, queue, lo, hi);
}
/**
* Returns the number of keys in the symbol table in the given range.
*
* @param lo minimum endpoint
* @param hi maximum endpoint
* @return the number of keys in the symbol table between {@code lo}
* (inclusive) and {@code hi} (inclusive)
* @throws NullPointerException if either {@code lo} or {@code hi}
* is {@code null}
*/
public int size(Key lo, Key hi) {
if (lo == null) throw new NullPointerException("first argument to size() is null");
if (hi == null) throw new NullPointerException("second argument to size() is null");
if (lo.compareTo(hi) > 0) return 0;
if (contains(hi)) return rank(hi) - rank(lo) + 1;
else return rank(hi) - rank(lo);
}
/**
* Returns the height of the BST (for debugging).
*
* @return the height of the BST (a 1-node tree has height 0)
*/
public int height() {
return height(root);
}
private int height(Node x) {
if (x == null) return -1;
return 1 + Math.max(height(x.left), height(x.right));
}
/**
* Returns the keys in the BST in level order (for debugging).
*
* @return the keys in the BST in level order traversal
*/
public Iterable<Key> levelOrder() {
Queue<Key> keys = new Queue<Key>();
Queue<Node> queue = new Queue<Node>();
queue.enqueue(root);
while (!queue.isEmpty()) {
Node x = queue.dequeue();
if (x == null) continue;
keys.enqueue(x.key);
queue.enqueue(x.left);
queue.enqueue(x.right);
}
return keys;
}
/*************************************************************************
* Check integrity of BST data structure.
***************************************************************************/
private boolean check() {
if (!isBST()) StdOut.println("Not in symmetric order");
if (!isSizeConsistent()) StdOut.println("Subtree counts not consistent");
if (!isRankConsistent()) StdOut.println("Ranks not consistent");
return isBST() && isSizeConsistent() && isRankConsistent();
}
// does this binary tree satisfy symmetric order?
// Note: this test also ensures that data structure is a binary tree since order is strict
private boolean isBST() {
return isBST(root, null, null);
}
// is the tree rooted at x a BST with all keys strictly between min and max
// (if min or max is null, treat as empty constraint)
// Credit: Bob Dondero's elegant solution
private boolean isBST(Node x, Key min, Key max) {
if (x == null) return true;
if (min != null && x.key.compareTo(min) <= 0) return false;
if (max != null && x.key.compareTo(max) >= 0) return false;
return isBST(x.left, min, x.key) && isBST(x.right, x.key, max);
}
// are the size fields correct?
private boolean isSizeConsistent() { return isSizeConsistent(root); }
private boolean isSizeConsistent(Node x) {
if (x == null) return true;
if (x.size != size(x.left) + size(x.right) + 1) return false;
return isSizeConsistent(x.left) && isSizeConsistent(x.right);
}
// check that ranks are consistent
private boolean isRankConsistent() {
for (int i = 0; i < size(); i++)
if (i != rank(select(i))) return false;
for (Key key : keys())
if (key.compareTo(select(rank(key))) != 0) return false;
return true;
}
/**
* Unit tests the {@code BST} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
BST<String, Integer> st = new BST<String, Integer>();
for (int i = 0; !StdIn.isEmpty(); i++) {
String key = StdIn.readString();
st.put(key, i);
}
for (String s : st.levelOrder())
StdOut.println(s + " " + st.get(s));
StdOut.println();
for (String s : st.keys())
StdOut.println(s + " " + st.get(s));
}
}
网友评论