Binary_search_tree from wikipedia

from: http://en.wikipedia.org/wiki/Binary_search_tree

http://zh.wikipedia.org/wiki/二進制搜尋樹

Binary search tree

From Wikipedia, the free encyclopedia

Binary Search Tree
Type	Tree
Time complexity in big O notation
Average	Worst case
Space	O(n)	O(n)
Search	O(log n)	O(n)
Insert	O(log n)	O(n)
Delete	O(log n)	O(n)

A binary search tree of size 9 and depth 3, with root 8 and leaves 1, 4, 7 and 13

In computer science, a binary search tree (BST), which may sometimes also be called an ordered or sorted binary tree, is a node-based binary tree data structurewhich has the following properties:[1]

• The left subtree of a node contains only nodes with keys less than the node's key.
• The right subtree of a node contains only nodes with keys greater than the node's key.
• Both the left and right subtrees must also be binary search trees.

Generally, the information represented by each node is a record rather than a single data element. However, for sequencing purposes, nodes are compared according to their keys rather than any part of their associated records.

The major advantage of binary search trees over other data structures is that the related sorting algorithms and search algorithms such as in-order traversal can be very efficient.

Binary search trees are a fundamental data structure used to construct more abstract data structures such as sets, multisets, and associative arrays.

Contents    [hide]  1 Operations 1.1 Searching 1.2 Insertion 1.3 Deletion 1.4 Traversal 1.5 Sort 2 Types 2.1 Performance comparisons 2.2 Optimal binary search trees 3 See also 4 References 5 Further reading 6 External links

[edit]Operations

Operations on a binary search tree require comparisons between nodes. These comparisons are made with calls to a comparator, which is a subroutine that computes the total order (linear order) on any two values. This comparator can be explicitly or implicitly defined, depending on the language in which the BST is implemented.

[edit]Searching

Searching a binary search tree for a specific value can be a recursive or iterative process. This explanation covers a recursive method.

We begin by examining the root node. If the tree is null, the value we are searching for does not exist in the tree. Otherwise, if the value equals the root, the search is successful. If the value is less than the root, search the left subtree. Similarly, if it is greater than the root, search the right subtree. This process is repeated until the value is found or the indicated subtree is null. If the searched value is not found before a null subtree is reached, then the item must not be present in the tree.

Here is the search algorithm in the Python programming language:

# 'node' refers to the parent-node in this case
 def search_binary_tree(node, key):
     if node is None:
         return None  # key not found
     if key < node.key:
         return search_binary_tree(node.leftChild, key)
     elif key > node.key:
         return search_binary_tree(node.rightChild, key)
     else:  # key is equal to node key
         return node.value  # found key
      

… or equivalent Haskell:

searchBinaryTree _   NullNode = Nothing
 searchBinaryTree key (Node nodeKey nodeValue (leftChild, rightChild)) =
     case compare key nodeKey of
       LT -> searchBinaryTree key leftChild
       GT -> searchBinaryTree key rightChild
       EQ -> Just nodeValue
      

This operation requires O(log n) time in the average case, but needs O(n) time in the worst case, when the unbalanced tree resembles a linked list (degenerate tree).

Assuming that BinarySearchTree is a class with a member function "search(int)" and a pointer to the root node, the algorithm is also easily implemented in terms of an iterative approach. The algorithm enters a loop, and decides whether to branch left or right depending on the value of the node at each parent node.

bool BinarySearchTree::search(int val)
{
    Node *next = this->root();
 
    while (next != NULL) {
        if (val == next->value()) {
            return true;
        } else if (val < next->value()) {
            next = next->left();   
        } else {
            next = next->right();
        }
    } 
 
    //not found
    return false;
}
      

[edit]Insertion

Insertion begins as a search would begin; if the root is not equal to the value, we search the left or right subtrees as before. Eventually, we will reach an external node and add the value as its right or left child, depending on the node's value. In other words, we examine the root and recursively insert the new node to the left subtree if the new value is less than the root, or the right subtree if the new value is greater than or equal to the root.

Here's how a typical binary search tree insertion might be performed in C++:

/* Inserts the node pointed to by "newNode" into the subtree rooted at "treeNode" */
 void InsertNode(Node* &treeNode, Node *newNode)
 {
     if (treeNode == NULL)
       treeNode = newNode;
     else if (newNode->key < treeNode->key)
       InsertNode(treeNode->left, newNode);
     else
       InsertNode(treeNode->right, newNode);
 }
      

The above "destructive" procedural variant modifies the tree in place. It uses only constant space, but the previous version of the tree is lost. Alternatively, as in the following Python example, we can reconstruct all ancestors of the inserted node; any reference to the original tree root remains valid, making the tree a persistent data structure:

def binary_tree_insert(node, key, value):
     if node is None:
         return TreeNode(None, key, value, None)
     if key == node.key:
         return TreeNode(node.left, key, value, node.right)
     if key < node.key:
         return TreeNode(binary_tree_insert(node.left, key, value), node.key, node.value, node.right)
     else:
         return TreeNode(node.left, node.key, node.value, binary_tree_insert(node.right, key, value))
      

The part that is rebuilt uses Θ(log n) space in the average case and O(n) in the worst case (see big-O notation).

In either version, this operation requires time proportional to the height of the tree in the worst case, which is O(log n) time in the average case over all trees, but O(n) time in the worst case.

Another way to explain insertion is that in order to insert a new node in the tree, its value is first compared with the value of the root. If its value is less than the root's, it is then compared with the value of the root's left child. If its value is greater, it is compared with the root's right child. This process continues, until the new node is compared with a leaf node, and then it is added as this node's right or left child, depending on its value.

There are other ways of inserting nodes into a binary tree, but this is the only way of inserting nodes at the leaves and at the same time preserving the BST structure.

Here is an iterative approach to inserting into a binary search tree in Java:

private Node m_root;
 
public void insert(int data) {
    if (m_root == null) {
        m_root = new TreeNode(data, null, null);
        return;
    }
    Node root = m_root;
    while (root != null) {
        // Not the same value twice
        if (data == root.getData()) {
            return;
        } else if (data < root.getData()) {
            // insert left
            if (root.getLeft() == null) {
                root.setLeft(new TreeNode(data, null, null));
                return;
            } else {
                root = root.getLeft();
            }
        } else {
            // insert right
            if (root.getRight() == null) {
                root.setRight(new TreeNode(data, null, null));
                return;
            } else {
                root = root.getRight();
            }
        }
    }
}
      

Below is a recursive approach to the insertion method.

private Node m_root;
 
public void insert(int data){
    if (m_root == null) {
        m_root = TreeNode(data, null, null);     
    }else{
        internalInsert(m_root, data);
    }
}
 
private static void internalInsert(Node node, int data){
    // Not the same value twice
    if (data == node.getValue()) {
        return;
    } else if (data < node.getValue()) {
        if (node.getLeft() == null) {
            node.setLeft(new TreeNode(data, null, null));
        }else{
            internalInsert(node.getLeft(), data);
        }
    }else{
        if (node.getRight() == null) {
            node.setRight(new TreeNode(data, null, null));
        }else{
            internalInsert(node.getRight(), data);
        } 
    }
}
      

[edit]Deletion

There are three possible cases to consider:

• Deleting a leaf (node with no children): Deleting a leaf is easy, as we can simply remove it from the tree.
• Deleting a node with one child: Remove the node and replace it with its child.
• Deleting a node with two children: Call the node to be deleted N. Do not delete N. Instead, choose either its in-order successor node or its in-order predecessor node, R. Replace the value of N with the value ofR, then delete R.

As with all binary trees, a node's in-order successor is the left-most child of its right subtree, and a node's in-order predecessor is the right-most child of its left subtree. In either case, this node will have zero or one children. Delete it according to one of the two simpler cases above.

Deleting a node with two children from a binary search tree. The triangles represent subtrees of arbitrary size, each with its leftmost and rightmost child nodes at the bottom two vertices.

Consistently using the in-order successor or the in-order predecessor for every instance of the two-child case can lead to an unbalanced tree, so good implementations add inconsistency to this selection.

Running Time Analysis: Although this operation does not always traverse the tree down to a leaf, this is always a possibility; thus in the worst case it requires time proportional to the height of the tree. It does not require more even when the node has two children, since it still follows a single path and does not visit any node twice.

Here is the code in Python:

def findMin(self):
    '''
    Finds the smallest element that is a child of *self*
    '''
    current_node = self
    while current_node.left_child:
        current_node = current_node.left_child
    return current_node
 
def replace_node_in_parent(self, new_value=None):
    '''
    Removes the reference to *self* from *self.parent* and replaces it with *new_value*.
    '''
    if self.parent:
        if self == self.parent.left_child:
            self.parent.left_child = new_value
        else:
            self.parent.right_child = new_value
    if new_value:
        new_value.parent = self.parent
 
def binary_tree_delete(self, key):
    if key < self.key:
        self.left_child.binary_tree_delete(key)
    elif key > self.key:
        self.right_child.binary_tree_delete(key)
    else: # delete the key here
        if self.left_child and self.right_child: # if both children are present
            # get the smallest node that's bigger than *self*
            successor = self.right_child.findMin()
            self.key = successor.key
            # if *successor* has a child, replace it with that
            # at this point, it can only have a *right_child*
            # if it has no children, *right_child* will be "None"
            successor.replace_node_in_parent(successor.right_child)
        elif self.left_child or self.right_child:   # if the node has only one child
            if self.left_child:
                self.replace_node_in_parent(self.left_child)
            else:
                self.replace_node_in_parent(self.right_child)
        else: # this node has no children
            self.replace_node_in_parent(None)
      

Source code in C++ (from http://www.algolist.net/Data_structures/Binary_search_tree). This URL also explains the operation nicely using diagrams.

bool BinarySearchTree::remove(int value) {
  if (root == NULL)
    return false;
  else {
    if (root->getValue() == value) {
      BSTNode auxRoot(0);
      auxRoot.setLeftChild(root);
      BSTNode* removedNode = root->remove(value, &auxRoot);
      root = auxRoot.getLeft();
      if (removedNode != NULL) {
        delete removedNode;
        return true;
      } else
        return false;
    } else {
      BSTNode* removedNode = root->remove(value, NULL);
      if (removedNode != NULL) {
        delete removedNode;
        return true;
      } else
        return false;
    }
  }
}
 
BSTNode* BSTNode::remove(int value, BSTNode *parent) {
  if (value < this->value) {
    if (left != NULL)
      return left->remove(value, this);
    else
      return NULL;
  } else if (value > this->value) {
    if (right != NULL)
      return right->remove(value, this);
    else
      return NULL;
  } else {
    if (left != NULL && right != NULL) {
      this->value = right->minValue();
      return right->remove(this->value, this);
    } else if (parent->left == this) {
      parent->left = (left != NULL) ? left : right;
      return this;
    } else if (parent->right == this) {
      parent->right = (left != NULL) ? left : right;
      return this;
    }
  }
}
 
int BSTNode::minValue() {
  if (left == NULL)
    return value;
  else
    return left->minValue();
}
      

[edit]Traversal

Main article:  Tree traversal

Once the binary search tree has been created, its elements can be retrieved in-order by recursively traversing the left subtree of the root node, accessing the node itself, then recursively traversing the right subtree of the node, continuing this pattern with each node in the tree as it's recursively accessed. As with all binary trees, one may conduct a pre-order traversal or a post-order traversal, but neither are likely to be useful for binary search trees.

The code for in-order traversal in Python is given below. It will call callback for every node in the tree.

def traverse_binary_tree(node, callback):
    if node is None:
        return
    traverse_binary_tree(node.leftChild, callback)
    callback(node.value)
    traverse_binary_tree(node.rightChild, callback)
      

Traversal requires Ω(n) time, since it must visit every node. This algorithm is also O(n), so it is asymptotically optimal.

Template:In Order Traversal in Binary Search Tree without Recursion

The Code for in-order traversal in Language C is given below.

void InOrderTraversal(struct Node *n)
{
        struct Node *Cur, *Pre;
        if(n==NULL)
                return;
 
        Cur = n;
        while(Cur != NULL)
        {
                if(Cur->lptr == NULL)
                {
                        printf("\t%d",Cur->val);
                        Cur= Cur->rptr;
                }
                else
                {
                        Pre = Cur->lptr;
                        while(Pre->rptr !=NULL && Pre->rptr != Cur)
                                Pre = Pre->rptr;
 
                        if (Pre->rptr == NULL)
                        {
                                Pre->rptr = Cur;
                                Cur = Cur->lptr;
                        }
                        else
                        {
                                Pre->rptr = NULL;
                                printf("\t%d",Cur->val);
                                Cur = Cur->rptr;
                        }
                }
        }
}
      

[edit]Sort

A binary search tree can be used to implement a simple but efficient sorting algorithm. Similar to heapsort, we insert all the values we wish to sort into a new ordered data structure—in this case a binary search tree—and then traverse it in order, building our result:

def build_binary_tree(values):
    tree = None
    for v in values:
        tree = binary_tree_insert(tree, v)
    return tree
 
def get_inorder_traversal(root):
    '''
    Returns a list containing all the values in the tree, starting at *root*.
    Traverses the tree in-order(leftChild, root, rightChild).
    '''
    result = []
    traverse_binary_tree(root, lambda element: result.append(element))
    return result
      

The worst-case time of 

build_binary_tree

 is O(n2)—if you feed it a sorted list of values, it chains them into a linked list with no left subtrees. For example, 

build_binary_tree([1, 2, 3, 4, 5])

 yields the tree 

(1 (2 (3 (4 (5)))))

.

There are several schemes for overcoming this flaw with simple binary trees; the most common is the self-balancing binary search tree. If this same procedure is done using such a tree, the overall worst-case time isO(nlog n), which is asymptotically optimal for a comparison sort. In practice, the poor cache performance and added overhead in time and space for a tree-based sort (particularly for node allocation) make it inferior to other asymptotically optimal sorts such as heapsort for static list sorting. On the other hand, it is one of the most efficient methods of incremental sorting, adding items to a list over time while keeping the list sorted at all times.

[edit]Types

There are many types of binary search trees. AVL trees and red-black trees are both forms of self-balancing binary search trees. A splay tree is a binary search tree that automatically moves frequently accessed elements nearer to the root. In a treap ("tree heap"), each node also holds a (randomly chosen) priority and the parent node has higher priority than its children. Tango Trees are trees optimized for fast searches.

Two other titles describing binary search trees are that of a complete and degenerate tree.

A complete tree is a tree with n levels, where for each level d <= n - 1, the number of existing nodes at level d is equal to 2d. This means all possible nodes exist at these levels. An additional requirement for a complete binary tree is that for the nth level, while every node does not have to exist, the nodes that do exist must fill from left to right.

A degenerate tree is a tree where for each parent node, there is only one associated child node. What this means is that in a performance measurement, the tree will essentially behave like a linked list data structure.

[edit]Performance comparisons

D. A. Heger (2004)[2] presented a performance comparison of binary search trees. Treap was found to have the best average performance, while red-black tree was found to have the smallest amount of performance fluctuations.

[edit]Optimal binary search trees

If we don't plan on modifying a search tree, and we know exactly how often each item will be accessed, we can construct an optimal binary search tree, which is a search tree where the average cost of looking up an item (the expected search cost) is minimized.

Even if we only have estimates of the search costs, such a system can considerably speed up lookups on average. For example, if you have a BST of English words used in a spell checker, you might balance the tree based on word frequency in text corpora, placing words like "the" near the root and words like "agerasia" near the leaves. Such a tree might be compared with Huffman trees, which similarly seek to place frequently-used items near the root in order to produce a dense information encoding; however, Huffman trees only store data elements in leaves and these elements need not be ordered.

If we do not know the sequence in which the elements in the tree will be accessed in advance, we can use splay trees which are asymptotically as good as any static search tree we can construct for any particular sequence of lookup operations.

Alphabetic trees are Huffman trees with the additional constraint on order, or, equivalently, search trees with the modification that all elements are stored in the leaves. Faster algorithms exist for optimal alphabetic binary trees (OABTs).

Example:

procedure Optimum Search Tree(f, f´, c):
 for j = 0 to n do
  c[j, j] = 0, F[j, j] = f´j
  for d = 1 to n do
   for i = 0 to (n − d) do
    j = i + d
    F[i, j] = F[i, j − 1] + f´ + f´j
    c[i, j] = MIN(i<k<=j){c[i, k − 1] + c[k, j]} + F[i, j]
           

[edit]See also

• Search tree
• Binary tree
• Binary search algorithm
• Self-balancing binary search tree
• Randomized binary search tree
• Red-black tree
• AA tree
• Tango Trees
• B-tree
• Data structure
• Trie
• Ternary search tree
• Elastic binary tree
• Hash table
• Skip list

[edit]References

1. ^ Gilberg, R.; Forouzan, B. (2001), "8", Data Structures: A Pseudocode Approach With C++, Pacific Grove, CA: Brooks/Cole, p. 339, ISBN 0-534-95216-X
2. ^ Heger, Dominique A. (2004), "A Disquisition on The Performance Behavior of Binary Search Tree Data Structures", European Journal for the Informatics Professional 5 (5): 67–75

[edit]Further reading

• Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89685-0. Section 6.2.2: Binary Tree Searching, pp. 426–458.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 12: Binary search trees, pp. 253–272. Section 15.5: Optimal binary search trees, pp. 356–363.

[edit]External links

• Binary Search Tree C++ and Pascal
• Binary Search Trees Animation
• Full source code to an efficient implementation in C++
• Implementation of a Persistent Binary Search Tree in C
• Iterative Implementation of Binary Search Trees in C#
• An introduction to binary trees from Stanford
• Dictionary of Algorithms and Data Structures - Binary Search Tree
• Binary Search Tree Example in Python
• Interactive Data Structure Visualizations - Binary Tree Traversals
• Literate implementations of binary search trees in various languages on LiteratePrograms
• BST Tree Applet by Kubo Kovac
• Well-illustrated explanation of binary search tree. Implementations in Java and C++
• Teacing Binary Search Tree through visualization
• Elastic Binary Trees (ebtree) - description and implementation in C.

二進制搜尋樹

維基百科，自由的百科全書

3層二進制搜尋樹

二叉查找樹（Binary Search Tree），或者是一棵空樹，或者是具有下列性質的二叉樹：

1. 若它的左子樹不空，則左子樹上所有結點的值均小于它的根結點的值；
2. 若它的右子樹不空，則右子樹上所有結點的值均大于它的根結點的值；
3. 它的左、右子樹也分别為二叉排序樹。

二叉排序樹的查找過程和次優二叉樹類似，通常采取二叉連結清單作為二叉排序樹的存儲結構。中序周遊二叉排序樹可得到一個關鍵字的有序序列，一個無序序列可以通過構造一棵二叉排序樹變成一個有序序列，構造樹的過程即為對無序序列進行排序的過程。每次插入的新的結點都是二叉排序樹上新的葉子結點，在進行插入操作時，不必移動其它結點，隻需改動某個結點的指針，由空變為非空即可。搜尋,插入,删除的複雜度等于樹高，期望O(logn),最壞O(n)(數列有序,樹退化成線性表).

雖然二叉排序樹的最壞效率是O(n),但它支援動态查詢,且有很多改進版的二叉排序樹可以使樹高為O(logn),如SBT,AVL,紅黑樹等.故不失為一種好的動态排序方法.

目錄    [隐藏]  1 二進制排序樹的查找算法 2 在二叉排序樹插入結點的算法 3 在二叉排序樹删除結點的算法 4 二叉排序樹性能分析 5 二叉排序樹的優化

[編輯]二進制排序樹的查找算法

在二進制排序樹b中查找x的過程為：

1. 若b是空樹，則搜尋失敗，否則：
2. 若x等於b的根節點的數據域之值，則查找成功；否則：
3. 若x小於b的根節點的數據域之值，則搜尋左子樹；否則：
4. 查找右子樹。

/* 以下代碼為C++寫成, 下同 */
Status SearchBST(BiTree T, KeyType key, BiTree f, BiTree &p){
  //在根指針T所指二進制排序樹中遞歸地查找其關鍵字等於key的數據元素，若查找成功，
  //則指針p指向該數據元素節點，并傳回TRUE，否則指針指向查找路徑上訪問的最後
  //一個節點并傳回FALSE，指針f指向T的雙親，其初始調用值為NULL
  if(!T) { //查找不成功
    p=f;
    return false;
  }
  else if (key == T->data.key) { //查找成功
    p=T;
    return true;
  }
  else if (key < T->data.key) //在左子樹中繼續查找
    return SearchBST(T->lchild, key, T, p);
  else //在右子樹中繼續查找
    return SearchBST(T->rchild, key, T, p);
}
      

[編輯]在二叉排序樹插入結點的算法

向一個二叉排序樹b中插入一個結點s的算法，過程為：

1. 若b是空樹，則将s所指結點作為根結點插入，否則：
2. 若s->data等于b的根結點的資料域之值，則傳回，否則：
3. 若s->data小于b的根結點的資料域之值，則把s所指結點插入到左子樹中，否則：
4. 把s所指結點插入到右子樹中。

/*當二叉排序樹T中不存在關鍵字等于e.key的資料元素時，插入e并傳回TRUE，否則傳回FALSE*/
Status InsertBST(BiTree &T, ElemType e){
  if(!SearchBST(T, e.key, NULL,p){
      s = new BiTNode;
      s->data = e; s->lchild = s->rchild = NULL;
      if(!p)  
        T=s;  //被插結點*s為新的根結點
      else if (e.key < p->data.key)
        p->lchld = s;    //被子插結點*s為左孩子
      else 
        p->rchild = s;   //被插結點*s為右孩子
      return true;
    }
    else 
      return false;  //樹中已有關鍵字相同的結點，不再插入
    }
}
      

[編輯]在二叉排序樹删除結點的算法

在二叉排序樹删去一個結點，分三種情況讨論：

1. 若*p結點為葉子結點，即PL(左子樹)和PR(右子樹)均為空樹。由于删去葉子結點不破壞整棵樹的結構，則隻需修改其雙親結點的指針即可。
2. 若*p結點隻有左子樹PL或右子樹PR，此時隻要令PL或PR直接成為其雙親結點*f的左子樹（當*p是左子樹）或右子樹（當*p是右子樹）即可，作此修改也不破壞二叉排序樹的特性。
3. 若*p結點的左子樹和右子樹均不空。在删去*p之後，為保持其它元素之間的相對位置不變，可按中序周遊保持有序進行調整，可以有兩種做法：其一是令*p的左子樹為*f的左子樹，*s為*f左子樹的最右下的結點，而*p的右子樹為*s的右子樹；其二是令*p的直接前驅（或直接後繼）替代*p，然後再從二叉排序樹中删去它的直接前驅（或直接後繼）。在二叉排序樹上删除一個結點的算法如下：

Status DeleteBST(BiTree &T, KeyType key){
  //若二叉排序樹T中存在關鍵字等于key的資料元素時，則删除該資料元素，并傳回
  //TRUE；否則傳回FALSE
  if(!T) 
    return false;    //不存在關鍵字等于key的資料元素
  else{
    if(key == T->data.key) {    //  找到關鍵字等于key的資料元素
      return Delete(T);
    }
    else if(key > T->data.key)
      return DeleteBST(T->lchild, key);
    else
      return DeleteBST(T->rchild, key);
  }
}
 
Status Delete(BiTree &p){
  //從二叉排序樹中删除結點p，并重接它的左或右子樹
  if(!p->rchild){        //右子樹空則隻需重接它的左子樹
    q=p;
    p=p->lchild;
    delete q;
  }
  else if(!p->lchild){  //左子樹空隻需重接它的右子樹
    q=p;
    p=p->rchild; 
    delete q;
  }
  else{  //左右子樹均不空
    q=p; 
    s=p->lchild;
    while(s->rchild){ 
      q=s; 
      s=s->rchild;
    }     //轉左，然後向右到盡頭
    p->data = s->data;      //s指向被删結點的“前驅”
    if(q!=p)       
      q->rchild = s->lchild;        //重接*q的右子樹
    else 
      q->lchild = s->lchild;        //重接*q的左子樹
    delete s;
  }
  return true;
}
      

[編輯]二叉排序樹性能分析

每個結點的Ci為該結點的層次數。最壞情況下，當先後插入的關鍵字有序時，構成的二叉排序樹蛻變為單支樹，樹的深度為n，其平均查找長度為

（和順序查找相同），最好的情況是二叉排序樹的形态和折半查找的判定樹相同，其平均查找長度和log 2(n)成正比（O(log 2(n))）。

[編輯]二叉排序樹的優化

請參見主條目平衡樹。

1. Size Balanced Tree(SBT)
2. AVL樹
3. 紅黑樹
4. Treap(Tree+Heap)

這些均可以使查找樹的高度為O(log(n))