depth：n的depth為從root到n的路徑的長(zhǎng)度
height：n的height為從n到一個(gè)leaf的最長(zhǎng)路徑長(zhǎng)度。樹的height等于root的depth

preorder traversal: 遍歷順序，根，左子樹，右子樹
postorder traversal: 遍歷順序，左子樹，右子樹，根
inorder traversal: 遍歷順序, 左子樹，根，右子樹

• Bineary search tree
  average depth = O(logN)
  worst case depth is O(N)
  implementation:

struct BinaryNode
{
    Object element; //The data in the node
    BinaryNode *left; //Left child
    BinaryNode *right; //Right child
};

contains:

/*Internam method to test if an item is in a subtree
  * x is itemto search for.
  * t is the node that roots the subtree
*/
bool contains(const Comparable & x, BinaryNode *t) const
{
    if (t == NULL) return false;
    else if (x < t->element) return contains(x, t->left);
    else if (x > t->element) return contains(x, t->right);
    else return true; //match
}

findMin and findMax (recursion and iteration implementation):

//recursion, return node containing the smallest item
BinaryNode * findMin(BinaryNode *t) const
{
    if(t == NULL) return NULL;
    if(t->left == NULL) return t;
    return findMin(t->left);
}
//iteration, return node containing the largest item
BinaryNode * findMax(BinaryNode *t) const
{
    if(t != NULL)
    {
        while (t->right != NULL)
        {
          t = t->right;
        }
    }
    return t;
}

insert:

/* x is the item to nsert
   t is the node that roots the subtree
   set the new root of the subtree
*/
void insert(const Comparable & x, BinaryNode * &t)
{
    if(t == NULL) t = new BinaryNode(x, NULL, NULL);
    else if (x < t->element) insert(x, t->left);
    else if (x > t->element) insert(x, t->right);
    else ; //duplicate, do nothing
}

remove:
需要考慮多種情況。
如果結(jié)點(diǎn)是樹葉，可以直接刪除。
如果結(jié)點(diǎn)有一個(gè)child，刪除后將child與父結(jié)點(diǎn)連接，如下

Paste_Image.png

如果結(jié)點(diǎn)有兩個(gè)child，一般的刪除策略是用其右字?jǐn)?shù)中的最小數(shù)據(jù)代替該節(jié)點(diǎn)并遞歸地刪除那個(gè)節(jié)點(diǎn)。例子如下

Paste_Image.png

//效率不高，因?yàn)檠貥鋬纱嗡阉鱽?lái)查找和刪除右子樹中的最小節(jié)點(diǎn)
//如要提高效率，需要寫一個(gè)特殊的removeMin  
void remove(const Comparable & x, BinaryNode * &t)
{
    if(t == NULL) return; //item not found, do nothing
    if(x < t->element) remove(x,t->left);
    else if(x > t->element) remove(x,t->right);
    else if(t->left != NULL && t->right != NULL) //Two children
    {
        t->element = findMin(t->right)->element;
        remove(t->element,t->right);
    } 
    else //node has only one child
    {
        BinaryNode *oldNode = t;
        t = (t->left != NULL) ? t->left : t->right;
        delete oldNode;
    }
}

makeEmpty and destructor:

~BinarySearchTree()
{
    makeEmpty();
}
void makeEmpty(BinaryNode * &t)
{
    if (t != NULL)
    {
        makeEmpty(t->left);
        makeEmpty(t->right);
        delete t;
    }
    t = NULL; //important
}

operator= and clone:

//deep copy
const BinarySearchTree & operator=(const BinarySearchTree &rhs)
{
    if(this != &rhs)
    {
        makeEmpty();
        root = clone(rhs.root);
    }
    return *this;
}
//method to clone subtree
BinaryNode *clone(BinaryNode *t) const
{
    if(t == NULL) return NULL;
    return new BinaryNode(t->element,clone(t->left),clone(t->right));
}

• Complexity analysis:
  average case:
  O(logN) for all operation
  actually O(d) for all operations except makeEmpty and operator=
  d = depth including the node visited
  worst case: O(N)

• Tree traversal:
  O(N) complexity
  print tree in sorted order:

//inorder
void printTree(BinaryNode *t) const
{
    if(t != NULL)
    {
        printTree(t->left);
        cout << t->element << endl;
        printTree(t->right);
    }
}

compute height:

int height(BinaryNode *t)
{
    if(t == NULL) return -1;
    else return 1 + max(height(t->left),height(t->right));
}