java二叉排序樹的概念和操作

一：概念
二叉搜索樹又稱二叉排序樹，它或者是一棵空樹，或者是具有以下性質(zhì)的二叉樹:
若它的左子樹不為空，則左子樹上所有節(jié)點(diǎn)的值都小于根節(jié)點(diǎn)的值
若它的右子樹不為空，則右子樹上所有節(jié)點(diǎn)的值都大于根節(jié)點(diǎn)的值
它的左右子樹也分別為二叉搜索樹。

二：操作——查找
先和根節(jié)點(diǎn)做對(duì)比，相等返回，如果不相等，
關(guān)鍵碼key>根節(jié)點(diǎn)key,在右子樹中找(root=root.rightChild)
關(guān)鍵碼key<根節(jié)點(diǎn)key,在左子樹中找(root=root.leftChild)
否則返回false

三：操作——插入
根據(jù)二叉排序樹的性質(zhì)，左孩子比根節(jié)點(diǎn)的值小，右孩子比根節(jié)點(diǎn)的值大。關(guān)鍵碼key先于根節(jié)點(diǎn)key作比較，然后再判斷與根節(jié)點(diǎn)的左或者右作比較，滿足二叉排序樹性質(zhì)時(shí)，即為合理位置，然后插入。

四： 操作-刪除（難點(diǎn)）
設(shè)待刪除結(jié)點(diǎn)為 cur, 待刪除結(jié)點(diǎn)的雙親結(jié)點(diǎn)為 parent
1. cur.left == null

1. cur 是 root，則 root = cur.right
2. cur 不是 root，cur 是 parent.left，則 parent.left = cur.right
3. cur 不是 root，cur 是 parent.right，則 parent.right = cur.right
   2. cur.right == null
4. cur 是 root，則 root = cur.left
5. cur 不是 root，cur 是 parent.left，則 parent.left = cur.left
6. cur 不是 root，cur 是 parent.right，則 parent.right = cur.left
   3. cur.left != null && cur.right != null
7. 需要使用替換法進(jìn)行刪除，即在它的右子樹中尋找中序下的第一個(gè)結(jié)點(diǎn)(關(guān)鍵碼最小)，用它的值填補(bǔ)到被刪除節(jié)點(diǎn)中，再來處理該結(jié)點(diǎn)的刪除問題

五：實(shí)現(xiàn)

public class BinarySearchTree<K extends Comparable<K>, V>
{ 
public static class Node<K extends Comparable<K>, V> 
{
K key; 
V value; 
Node<K, V> left;
Node<K, V> right;

public String toString()
{ 
return String.format("{%s, %s}", key, value);
}
}
private Node<K, V> root = null;
public V get(K key) 
{ 
Node<K, V> parent = null; 
Node<K, V> cur = root; 
while (cur != null)
{ 
parent = cur;
int r = key.compareTo(cur.key);
if (r == 0)
{ 
return cur.value;
} 
else if (r < 0) {
cur = cur.left; 
}

else 
{
cur = cur.right;
} 
}
return null; 
}
public V put(K key, V value)

{ 
if (root == null)
{ root = new Node<>();
root.key = key;
root.v
display(root);
return null;
}
Node<K, V> parent = null; 
Node<K, V> cur = root; 
while (cur != null) 
{ 
parent = cur;

int r = key.compareTo(cur.key);
if (r == 0) 
{ 
V oldValue = cur.value; 
cur.value = value; 
display(root); 
return oldValue; 
}
else if (r < 0)
{ 
cur = cur.left; 
} 
else

{ 
cur = cur.right;
} 
}
Node<K, V> node = new Node<>(); 
node.key = key; 
node.value = value;
int r = key.compareTo(parent.key);
if (r < 0)
{ parent.left = node;
} 
else { parent.right = node; 
}
display(root); 
return null; 
}
public V remove(K key) 
{ 

Node<K, V> parent = null; 
Node<K, V> cur = root; 
while (cur != null) 
{ 
int r = key.compareTo(cur.key);
if (r == 0)
{ 
V oldValue = cur.value; 
deleteNode(parent, cur);
display(root); 
return oldValue; } 
else if (r < 0)
{ parent = cur; cur = cur.left; }
else { parent = cur; cur = cur.right; 
} 
}
display(root); 
return null;
}
private void deleteNode(Node<K,V> parent, Node<K,V> cur) 
{
if (cur.left == null)
{
if (cur == root)
{
root = cur.right;
} 
else if (cur == parent.left)
{ parent.left = cur.right; }
else { parent.right = cur.right; }
} else if (cur.right == null)
{ 
if (cur == root)
{ root = cur.left; }
else if (cur == parent.left)
{ parent.left = cur.left; }
else { parent.right = cur.left; }
} else {
// 去 cur 的右子樹中尋找最小的 key 所在的結(jié)點(diǎn) scapegoat
// 即 scapegoat.left == null 的結(jié)點(diǎn)
Node<K,V> goatParent = cur;
Node<K,V> scapegoat = cur.right;
while (scapegoat.left != null)
{ goatParent = scapegoat; scapegoat = cur.left; }
cur.key = scapegoat.key;
cur.value = scapegoat.value;
if (scapegoat == goatParent.left)
{
goatParent.left = scapegoat.right;
}
else { goatParent.right = scapegoat.right; }
} 
}
private static <K extends Comparable<K>,V> void display(Node<K,V> node) 
{
System.out.print("前序: ");
preOrder(node);

System.out.println();
System.out.print("中序: ")
inOrder(node); 
System.out.println(); }
private static <K extends Comparable<K>,V> void preOrder(Node<K,V> node)
{ if (node == null)
{ return; }
System.out.print(node + " ");
preOrder(node.left);
preOrder(node.right); }
private static <K extends Comparable<K>,V> void inOrder(Node<K,V> node)
{ if (node == null) 
{ return; }
inOrder(node.left);
System.out.print(node + " ");
inOrder(node.right); }
public static void main(String[] args)
{ 
BinarySearchTree<Integer, String> tree = new BinarySearchTree<>(); 
int[] keys = { 5, 3, 7, 4, 2, 6, 1, 9, 8 }; 
for (int key : keys) 
{
tree.put(key, String.valueOf(key)); }
System.out.println("=================================="); tree.put(3, "修改過的 3"); System.out.println("=================================="); tree.remove(9);
tree.remove(1); t
ree.remove(3);
``` } 
}

**六：性能分析**
插入和刪除操作都必須先查找，查找效率代表了二叉搜索樹中各個(gè)操作的性能。
對(duì)有n個(gè)結(jié)點(diǎn)的二叉搜索樹，若每個(gè)元素查找的概率相等，則二叉搜索樹平均查找長(zhǎng)度是結(jié)點(diǎn)在二叉搜索樹的深度的函數(shù)，即結(jié)點(diǎn)越深，則比較次數(shù)越多。
但對(duì)于同一個(gè)關(guān)鍵碼集合，如果各關(guān)鍵碼插入的次序不同，可能得到不同結(jié)構(gòu)的二叉搜索樹。
**七： 和 java 類集的關(guān)系**
TreeMap 和 TreeSet 即 java 中利用搜索樹實(shí)現(xiàn)的 Map 和 Set；