Chapter 10
Trees 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
data
child1
Child2 …
data
first child
Next sibling
• first child and next sibling links
• Correspondence with binary trees
Conversion
(from forest/Orchard to binary tree
Corresponded binary tree
Corresponded binary tree
respectively
Huffman Tree(哈夫曼树)
Path Length (路径长度 )
Path Length of the binary tree
Total length of all from leaves to root
Weighted Path Length (WPL,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
与该结点到根的路径长度的乘积的和。
n 1
WPL   w i  l i
i 0
How about the WPLof the following
binary trees
Huffman tree
A (extended) binary tree with minimal
WPL。
Processes of huffman tree
Huffman coding
Compression
suppose we have a message：
CAST CAST SAT AT A TASA
alphabet ={ C, A, S, T }，frequency of them (次
数) are W＝{ 2, 7, 4, 5 }。
first case equal length coding
A : 00 T : 10 C : 01 S : 11
Total coding length of the message is
( 2+7+4+5 ) * 2 = 36.
Huffman tree ??
A : 0 T : 10
C : 110 S : 111
Total length of huffman coding：
7*1+5*2+( 2+4 )*3 = 35
Which 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。
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 empty
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
inorder, 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 Trees
Definition:
An AVL tree is a binary search tree in which the heights
of
the left and right subtrees of the root differ by at most 1
and
in which the left and right subtrees are again AVL trees.
With each node of an AVL tree is associated a balance
factor that is left higher, equal, or right higher according,
respectively, as the left subtree has height greater than,
equal
to, 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-Tree
3. 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