Chapter 10Trees and Binary Trees

Part 2

?Traversal level by level

Definitions

Rooted trees with four vertices(Root is at the top of tree.)

Ordered trees with four vertices

Implementations of Ordered Trees

• Multiple links

• first child and next sibling links• Correspondence with binary trees

data child1 Child2 …

data first child Next sibling

Conversion Conversion

(from forest/Orchard to binary tree(from forest/Orchard to binary tree

Corresponded binary tree respectively

Corresponded binary tree

Huffman Tree(Huffman Tree( 哈夫曼树哈夫曼树 ))

Path Length (Path Length ( 路径长度 路径长度 ))

Path Length of the binary tree Path Length of the binary tree Total length of all from leaves to rootTotal length of all from leaves to root

Weighted Path LengthWeighted Path Length ( (WPL,WPL, 带权路径长度 带权路径长度 ))

树的带权路径长度是树的各叶结点所带的权值树的带权路径长度是树的各叶结点所带的权值与该结点到根的路径长度的乘积的和。与该结点到根的路径长度的乘积的和。

1

0

n

iii lwWPL

How about the WPLof the following binary trees

Huffman tree

A (extended) binary tree with minimal A (extended) binary tree with minimal WPLWPL 。。

Processes of huffman treeProcesses of huffman tree

Huffman coding Huffman coding

Compression Compression suppose we have a messagesuppose we have a message ： ： CAST CAST SAT AT A TASACAST CAST SAT AT A TASA alphabet ={ C, A, S, T }alphabet ={ C, A, S, T } ，， frequency of them frequency of them (( 次数次数 ) are ) are WW＝＝ { 2, 7, 4, 5 }{ 2, 7, 4, 5 } 。。 first case equal length codingequal length coding AA : 00 T : 10 C : 01 S : 11 : 00 T : 10 C : 01 S : 11Total coding length of the message is Total coding length of the message is ( 2+7+4+5 ) * 2 = 36.( 2+7+4+5 ) * 2 = 36.

AA : 0 T : 10 C : 110 S : 111 : 0 T : 10 C : 110 S : 111Total length of huffman codingTotal length of huffman coding ： ：

7*1+5*2+7*1+5*2+( 2+4 )*3 = 35( 2+4 )*3 = 35Which is shorter than that of equal length codingWhich is shorter than that of equal length coding 。。霍夫曼编码是一种无前缀编码。解码时不会混淆。霍夫曼编码是一种无前缀编码。解码时不会混淆。

Huffman tree ??

Binary Search Trees• Can we find an implementation for

ordered lists in which we can search quickly (as with binary search on a contiguous list) and in which we can make insertions and deletions quickly (as with a linked list)?

DEFINITION• A binary search tree is a binary tree that is either empt

y or in which the data entry of every node has a key and satisfies the conditions:

1. The key of the left child of a node (if it exists) is less than the key of its parent node.

2. The key of the right child of a node (if it exists) is greater than the key of its parent node.

3. The left and right subtrees of the root are again binary search trees.We always require:No two entries in a binary search tree may have equal keys.

different views• We can regard binary search trees

as a new ADT.• We may regard binary search trees

as a specialization of binary trees.• We may study binary search trees

as a new implementation of the ADT ordered list.

The Binary Search Tree Class

Recursive auxiliary function:

template <class Record>Binary node<Record> *Search tree<Record> :: search for

node( Binary node<Record>* sub root, const Record &target) const

{if (sub root == NULL || sub root->data == target)return sub root;else if (sub root->data < target)return search for node(sub root->right, target);else return search for node(sub root->left, target);}

Nonrecursive version:template <class Record>Binary node<Record> *Search tree<Record> :: search for

node( Binary node<Record> *sub root, const Record &target) const

{while (sub root != NULL && sub root->data != target)if (sub root->data < target) sub root = sub root->right;else sub root = sub root->left;return sub root;}

Public method for tree search:template <class Record>

Error code Search tree<Record> ::tree search(Record &target) const

{Error code result = success;Binary node<Record> *found = search for node(root, target);if (found == NULL)

result = not present;else

target = found->data;return result;}

Binary Search Trees with the Same Keys

search

Creating a BST by insertion

insertion

Method for Insertion

Method for Insertion

Analysis of insertion

Treesort• When a binary search tree is traversed in in

order, the keys will come out in sorted order.

• This is the basis for a sorting method, called treesort: Take the entries to be sorted, use the method insert to build them into a binary search tree, and then use inorder traversal to put them out in order.

Treesort

Comparison First advantage of treesort over quicksort: The

nodes need not all be available at the start of the process, but are built into the tree one by one as they become available.

Second advantage: The search tree remains available for later insertions and removals.

Drawback: If the keys are already sorted, then treesort will be a disasteróthe search tree it builds will reduce to a chain. Treesort should never be used if the keys are already sorted, or are nearly so.

Removal from a Binary Search Tree

Removal from a Binary Search Tree (continue)

Height Balance: AVL TreesDefinition:An AVL tree is a binary search tree in which the heights ofthe left and right subtrees of the root differ by at most 1 andin which the left and right subtrees are again AVL trees.With each node of an AVL tree is associated a balancefactor that is left higher, equal, or right higher according,respectively, as the left subtree has height greater than, equ

alto, or less than that of the right subtree.

Example AVL trees

Example AVL trees

Example non-AVL trees

Example non-AVL trees

Insertions into an AVL tree

Insertions into an AVL tree

Rotations of an AVL Tree

Double Rotation

Deletion with no rotations

Deletion with single left rotations

Deletion with double rotation

Worst-Case AVL Trees

Fibonacci Trees

Analysis of Fibonacci Trees

Analysis of Fibonacci Trees

Multiway Search Trees• An m-way search tree is a tree in

which, for some integer m called the order of the tree, each node has at most m children.

Balanced Multiway Trees (B-Trees)

B-Tree Example

Insertion into a B-Tree

In contrast to binary search trees, B-trees are not allowed to grow at their leaves; instead, they are forced to grow at the root. General insertion method:

1. Search the tree for the new key. This search (if the key is truly new) will terminate in failure at a leaf.

2. Insert the new key into to the leaf node. If the node was not previously full, then the insertion is finished.

Insertion into a B-Tree3. When a key is added to a full node, then the

node splits into two nodes, side by side on the same level, except that the median key is not put into either of the two new nodes.

4. When a node splits, move up one level, insert the median key into this parent node, and repeat the splitting process if necessary.

5. When a key is added to a full root, then the root splits in two and the median key sent upward becomes a new root. This is the only time when the B-tree grows in height.

Growth of a B-Tree

Growth of a B-Tree

Growth of a B-Tree

Growth of a B-Tree

Deletion from a B-Tree

Deletion from a B-Tree

Deletion from a B-Tree