二叉树的每个结点至多只有二棵子树(不存在度大于2的结点)，二叉树的子树有左右之分，次序不能颠倒。二叉树的第i层至多有2^{i-1}个结点；深度为k的二叉树至多有2^k-1个结点；对任何一棵二叉树T，如果其终端结点数为n_0，度为2的结点数为n_2，则n_0=n_2+1。
一棵深度为k，且有2^k-1个节点称之为满二叉树；深度为k，有n个节点的二叉树，当且仅当其每一个节点都与深度为k的满二叉树中，序号为1至n的节点对应时，称之为完全二叉树。
<?php
/** * 二叉树的定义 */
class BinaryTree {
protected $key = NULL; // 当前节点的值
protected $left = NULL; // 左子树
protected $right = NULL; // 右子树
/** * 以指定的值构造二叉树，并指定左右子树 *
* @param mixed $key 节点的值.
* @param mixed $left 左子树节点.
* @param mixed $right 右子树节点.
*/
public function __construct( $key = NULL, $left = NULL, $right = NULL) {
$this->key = $key;
if ($key === NULL) {
$this->left = NULL;
$this->right = NULL;
}
elseif ($left === NULL) {
$this->left = new BinaryTree();
$this->right = new BinaryTree();
}
else {
$this->left = $left;
$this->right = $right;
}
}
/**
* 析构方法.
*/
public function __destruct() {
$this->key = NULL;
$this->left = NULL;
$this->right = NULL;
}
/**
* 清空二叉树.
**/
public function purge () {
$this->key = NULL;
$this->left = NULL;
$this->right = NULL;
}
/**
* 测试当前节点是否是叶节点.
*
* @return boolean 如果节点非空并且有两个空的子树时为真，否则为假.
*/
public function isLeaf() {
return !$this->isEmpty() &&
$this->left->isEmpty() &&
$this->right->isEmpty();
}
/**
* 测试节点是否为空
*
* @return boolean 如果节点为空返回真，否则为假.
*/
public function isEmpty() {
return $this->key === NULL;
}
/**
* Key getter.
*
* @return mixed 节点的值.
*/
public function getKey() {
if ($this->isEmpty()) {
return false;
}
return $this->key;
}
/**
* 给节点指定Key值,节点必须为空
*
* @param mixed $object 添加的Key值.
*/
public function attachKey($obj) {
if (!$this->isEmpty())
return false;
$this->key = $obj;
$this->left = new BinaryTree();
$this->right = new BinaryTree();
}
/**
* 删除key值，使得节点为空.
*/
public function detachKey() {
if (!$this->isLeaf())
return false;
$result = $this->key;
$this->key = NULL;
$this->left = NULL;
$this->right = NULL;
return $result;
}
/**
* 返回左子树
*
* @return object BinaryTree 当前节点的左子树.
*/
public function getLeft() {
if ($this->isEmpty())
return false;
return $this->left;
}
/**
* 给当前结点添加左子树
*
* @param object BinaryTree $t 给当前节点添加的子树.
*/
public function attachLeft(BinaryTree $t) {
if ($this->isEmpty() || !$this->left->isEmpty())
return false;
$this->left = $t;
}
/**
* 删除左子树
*
* @return object BinaryTree 返回删除的左子树.
*/
public function detachLeft() {
if ($this->isEmpty())
return false;
$result = $this->left;
$this->left = new BinaryTree();
return $result;
}
/**
* 返回当前节点的右子树
*
* @return object BinaryTree 当前节点的右子树.
*/
public function getRight() {
if ($this->isEmpty())
return false;
return $this->right;
}
/**
* 给当前节点添加右子树
*
* @param object BinaryTree $t 需要添加的右子树.
*/
public function attachRight(BinaryTree $t) {
if ($this->isEmpty() || !$this->right->isEmpty())
return false;
$this->right = $t;
}
/**
* 删除右子树，并返回此右子树
* @return object BinaryTree 删除的右子树.
*/
public function detachRight() {
if ($this->isEmpty ())
return false;
$result = $this->right;
$this->right = new BinaryTree();
return $result;
}
/**
* 先序遍历
*/
public function preorderTraversal() {
if ($this->isEmpty()) {
return ;
}
echo ' ', $this->getKey();
$this->getLeft()->preorderTraversal();
$this->getRight()->preorderTraversal();
}
/**
* 中序遍历
*/
public function inorderTraversal() {
if ($this->isEmpty()) {
return ;
}
$this->getLeft()->preorderTraversal();
echo ' ', $this->getKey();
$this->getRight()->preorderTraversal();
}
/**
* 后序遍历
*/
public function postorderTraversal() {
if ($this->isEmpty()) {
return ;
}
$this->getLeft()->preorderTraversal();
$this->getRight()->preorderTraversal();
echo ' ', $this->getKey();
}
}
/**
* 二叉排序树的PHP实现
*/
class BST extends BinaryTree {
/**
* 构造空的二叉排序树
*/
public function __construct() {
parent::__construct(NULL, NULL, NULL);
}
/**
* 析构
*/
public function __destruct() {
parent::__destruct();
}
/**
* 测试二叉排序树中是否包含参数所指定的值
*
* @param mixed $obj 查找的值.
* @return boolean True 如果存在于二叉排序树中则返回真，否则为假期
*/
public function contains($obj) {
if ($this->isEmpty())
return false;
$diff = $this->compare($obj);
if ($diff == 0) {
return true;
}elseif ($diff < 0) return $this->getLeft()->contains($obj);
else
return $this->getRight()->contains($obj);
}
/**
* 查找二叉排序树中参数所指定的值的位置
*
* @param mixed $obj 查找的值.
* @return boolean True 如果存在则返回包含此值的对象，否则为NULL
*/
public function find($obj) {
if ($this->isEmpty())
return NULL;
$diff = $this->compare($obj);
if ($diff == 0)
return $this->getKey();
elseif ($diff < 0) return $this->getLeft()->find($obj);
else
return $this->getRight()->find($obj);
}
/**
* 返回二叉排序树中的最小值
* @return mixed 如果存在则返回最小值，否则返回NULL
*/
public function findMin() {
if ($this->isEmpty ())
return NULL;
elseif ($this->getLeft()->isEmpty())
return $this->getKey();
else
return $this->getLeft()->findMin();
}
/**
* 返回二叉排序树中的最大值
* @return mixed 如果存在则返回最大值，否则返回NULL
*/
public function findMax() {
if ($this->isEmpty ())
return NULL;
elseif ($this->getRight()->isEmpty())
return $this->getKey();
else
return $this->getRight()->findMax();
}
/**
* 给二叉排序树插入指定值
*
* @param mixed $obj 需要插入的值.
* 如果指定的值在树中存在，则返回错误
*/
public function insert($obj) {
if ($this->isEmpty()) {
$this->attachKey($obj);
} else {
$diff = $this->compare($obj);
if ($diff == 0)
die('argu error');
if ($diff < 0) $this->getLeft()->insert($obj);
else
$this->getRight()->insert($obj);
}
$this->balance();
}
/**
* 从二叉排序树中删除指定的值
*
* @param mixed $obj 需要删除的值.
*/
public function delete($obj) {
if ($this->isEmpty ())
die();
$diff = $this->compare($obj);
if ($diff == 0) {
if (!$this->getLeft()->isEmpty()) {
$max = $this->getLeft()->findMax();
$this->key = $max;
$this->getLeft()->delete($max);
}
elseif (!$this->getRight()->isEmpty()) {
$min = $this->getRight()->findMin();
$this->key = $min;
$this->getRight()->delete($min);
} else
$this->detachKey();
} else if ($diff < 0) $this->getLeft()->delete($obj);
else
$this->getRight()->delete($obj);
$this->balance();
}
public function compare($obj) {
return $obj - $this->getKey();
}
/**
* Attaches the specified object as the key of this node.
* The node must be initially empty.
*
* @param object IObject $obj The key to attach.
* @exception IllegalOperationException If this node is not empty.
*/
public function attachKey($obj) {
if (!$this->isEmpty())
return false;
$this->key = $obj;
$this->left = new BST();
$this->right = new BST();
}
/**
* Balances this node.
* Does nothing in this class.
*/
protected function balance () {}
/**
* Main program.
*
* @param array $args Command-line arguments.
* @return integer Zero on success; non-zero on failure.
*/
public static function main($args) {
printf("BinarySearchTree main program.
");
$root = new BST();
foreach ($args as $row) {
$root->insert($row);
}
return $root;
}
}
$root = BST::main(array(50, 3, 10, 5, 100, 56, 78));
echo $root->findMax();
$root->delete(100);
echo $root->findMax();