Transcript Searching: Binary Trees Dr. Yingwu Zhu

Searching: Binary Trees
Dr. Yingwu Zhu
Review of Linear Search
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Collection of data items to be searched
is organized in a list
x1, x2, … xn
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Assume == and < operators defined for
the type
Linear search begins with item 1
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continue through the list until target found
or reach end of list
Linear Search
Vector based search function
template <typename t>
void LinearSearch (const vector<t>& v,
const t &item,
boolean &found, int &loc)
{
found = false; loc = 0;
for ( ; ; )
{
if (found || loc == v.size()) return;
if (item == v[loc]) found = true;
else loc++;
}
}
Linear Search
Singly-linked list based search function
template <typename t>
void LinearSearch (NodePointer first,
const t &item, bool &found, int
&loc)
{
NodePointer locptr=first;
for{loc=0; !found && locptr!=NULL;
locptr=locptr->next)
{
if (item == locptr->data)
found = true;
else loc++;
}
}
Linear Search
Iterator-based linear search
template <typename t>
void LinearSearch (const vector<t>& v ,
const t &item, bool &found, int &loc)
{ found = false; loc = 0;
for (vector<t>::iterator it=v.begin();
!found && it != v.end(); it++)
{
if (*it == item) found = true;
else loc++;
}
}
Binary Search
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Two requirements?
Binary Search
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Two requirements
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The data items are in ascending order (can
they be in decreasing order?? )
Direct access of each data item for
efficiency (why linked-list is not good!)
Binary Search
Binary search function for vector
template <typename t>
void LinearSearch (const vector<t> &v,
const t &item,
bool &found, int &loc) {
found = false; loc = 0;
int first = 0;
last = v.size() - 1;
for ( ; ; ){
if (found || first > last) return;
loc = (first + last) / 2;
if (item < v[loc])
last = loc - 1;
else if (item > v[loc])
first = loc + 1;
else
found = true; // item found
}
}
Binary Search vs. Linear Search
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Usually outperforms Linear search
O(logn) vs. O(n)
Disadvantages
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Sorted list of data items
Direct access of storage structure, not
good for linked-list
Good news: It is possible to use a
linked structure which can be searched
in a binary-like manner
Binary Search Tree
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Consider the following ordered list of integers
13
1.
2.
3.
28
35
49
62
66
80
Examine middle element
Examine left, right sublist (maintain pointers)
(Recursively) examine left, right sublists
Binary Search Tree
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Redraw the previous structure so that it has
a treelike shape – a binary tree
Trees
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A data structure which consists of
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a finite set of elements called nodes or vertices
a finite set of directed arcs which connect the
nodes
If the tree is nonempty
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one of the nodes (the root) has no incoming arc
every other node can be reached by following a
unique sequence of consecutive arcs (or paths)
Trees
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Tree terminology
Root node
• Children of the parent (3)
Leaf nodes
• Siblings to each other
Binary Trees
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Each node has at most two children
Useful in modeling processes where
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a comparison or experiment has exactly two
possible outcomes
the test is performed repeatedly
Example
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multiple coin tosses
encoding/decoding messages in dots and dashes
such as Mores code
Array Representation of Binary
Trees
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Store the ith node in the ith location of the
array
Array Representation of Binary
Trees
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Works OK for complete trees, not for sparse trees
Some Tree Definition, p656
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Complete trees
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Balanced trees
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Each level is completely filled except the bottom
level
The leftmost positions are filled at the bottom
level
Array storage is perfect for them
Binary trees
|left_subtree – right_subtree|<=1
Tree Height/Depth: # of levels
Tree Question?
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A complete tree must be a balanced
tree?
Give a node with position i in a
complete tree, what are the positions of
its child nodes?
Linked Representation of Binary
Trees
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Uses space more efficiently
Provides additional flexibility
Each node has two links
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one to the left child of the node
one to the right child of the node
if no child node exists for a node, the link is
set to NULL
Linked Representation of Binary
Trees
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Example
Binary Trees as Recursive Data
Structures
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A binary tree is either empty …
or
Consists of
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a node called the root
root has pointers to two
disjoint binary (sub)trees called …
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right (sub)tree
left (sub)tree
Which is either empty …
or …
Which is either empty …
or …
Anchor
Inductive
step
Tree Traversal is Recursive
If the binary tree is empty then
do nothing
Else
N: Visit the root, process data
L: Traverse the left subtree
R: Traverse the right subtree
The "anchor"
The inductive step
Traversal Order
Three possibilities for inductive step …
 Left subtree, Node, Right subtree
the inorder traversal
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Node, Left subtree, Right subtree
the preorder traversal
Left subtree, Right subtree, Node
the postorder traversal
ADT Binary Search Tree (BST)
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Collection of Data Elements
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binary tree
BST property: for each node x,
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value in left child of x < value in x < in right
child of x
Basic operations
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Construct an empty BST
Determine if BST is empty
Search BST for given item
ADT Binary Search Tree (BST)
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Basic operations (ctd)
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Insert a new item in the BST
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Delete an item from the BST
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Maintain the BST property
Maintain the BST property
Traverse the BST
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Visit each node exactly once
The inorder traversal must visit the values in
the nodes in ascending order
BST
template <typename T>
class BST {
public:
BST() : myRoot(0) {}
bool empty() { return myRoot==NULL; }
private:
class BinNode {
public:
T data;
BinNode* left, right;
BinNode() : left(0), right(0) {}
BinNode(T item): data(item), left(0), right(0) {}
}; //end of BinNode
typedef BinNode* BinNodePtr;
BinNodePtr myRoot;
};
BST operations
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Inorder, preorder and postorder
traverasl (recursive)
Non-recursive traversal
BST Traversals, p.670
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Why do we need two functions?
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Public: void inorder(ostream& out)
Private: inorderAux(…)
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Do the actual job
To the left subtree and then right subtree
Can we just use one function?
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Probably NO
Non-recursive traversals
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Let’s try non-recusive inorder
Any idea to implement non-recursive
traversals? What ADT can be used as an
aid tool?
Non-recursive traversals
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Basic idea
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Use LIFO data structure: stack
#include <stack> (provided by STL)
Chapter 9, google stack in C++ for more
details, members and functions
Useful in your advanced programming
Non-recursive inorder
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Let’s see how it works given a tree
Step by step
Non-recursive inorder
1. Set current node pointer ptr = myRoot
2. Push current node pointer ptr and all the leftmost nodes on its
subtree into a stack until meeting a NULL pointer
3. Check if the stack is empty or not. If yes, goto 5, otherwise goto
4
4. Take the top element in stack and assign it to ptr, print our ptr>data, pop the top element, and set ptr = ptr->right (visit
right subtree), goto 2
5. Done
Non-recursive inorder
template <typename DataType>
void non_recur_inorder(ostream& out) {
if (!myRoot) return; // empty BST
BinNodePointer ptr = myRoot;
stack<BST<DataType>::BinNodePointer> s; // empty stack
for ( ; ; ) {
while (ptr) { s.push(ptr);
ptr = ptr->left; } // leftmost nodes
if (s.empty()) break; // done
ptr = s.top();
s.pop();
out << ptr->data << “ “;
ptr = ptr->right; // right substree
}
}
Non-recursive traversals
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Can you do it yourself?
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Preorder?
Postorder?
BST Searches
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Search begins at root
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If that is desired item, done
If item is less, move down
left subtree
If item searched for is greater, move down right
subtree
If item is not found, we
will run into an empty subtree
BST Searches
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Write recursive search
Write Non-recursive search algorithm
bool search(const T& item) const
Insertion Operation
Basic idea:
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Use search operation to locate the insertion
position or already existing item
Use a parent point pointing to the parent of the
node currently being examined as descending
the tree
Insertion Operation
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Non-recursive insertion
Recursive insertion
Go through insertion illustration p.677
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Initially parent = NULL, locptr = root;
Location termination at NULL pointer or
encountering the item
Parent->left/right = item
Recursive Insertion
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See p. 681
Thinking!
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Why using & at subtreeroot
Deletion Operation
Three possible cases to delete a node, x,
from a BST
1. The node,
x, is a leaf
Deletion Operation
2. The node, x has one child
Deletion Operation
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x has two children
Delete node pointed to by
xSucc as described for
cases 1 and 2
Replace contents of x with
inorder successor
K
View
remove()
function
Question?
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Why do we choose inorder predecessor/successor to
replace the node to be deleted in case 3?
Implement Delete Operation
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void delete(const T& item) //remove the specified item if it exists
Assume you have this function
template <typename T>
void BST<T>::search2(const T& item, bool& found,
BST<T>::BinNodePtr& locptr, BST<T>::BinNodePtr& parent) const {
locptr = myRoot;
parent = NULL; found = false;
while(!found && locptr) {
if (item < locptr->data) {
parent = locptr;
locptr = locptr->left;
} else if (item > locptr->data) {
parent = locptr;
locptr = loc->right;
} else
found = true;
} //end while
}
Implement Delete Operation
1.
2.
3.
4.
5.
Search the specified item on the tree. If not found, then return;
otherwise go to 2.
Check if the node to be deleted has two child nodes (case 3). If so,
find the inorder successor/predecessor, replace the data, and then
make the successor/predecessor node the one to be deleted
In order to delete the node, you should keep track of its parent node
in step 1 and 2.
Delete the node according to the first two simple cases. Be careful
with parent
Release the memory of the node deleted
delete(const T& item)
{ BinNodePtr parent, x;
bool found;
search2(item, found, x, parent); //search item on the tree
if (!found)
return;
if (x->left && x->right) { //the case 3 with two child nodes
BinNodePtr xsucc =x->right;
parent = x;
while (xsucc->left) { //find inorder successor
parent = xsucc; xsucc = xsucc->left; }
x->data = xsucc->data;
x = xsucc;
} //end if
BinNodePtr subtree = x->left;
if (!subtree) subtree = x->right;
if (!parent)
myRoot = subtree;
else {
if (x == parent->left)
parent->left = subtree;
else
parent->right = subtree;
}
delete x;
}
BST Class Template
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View complete binary search tree
template, Fig. 12.7
View test program for BST, Fig. 12.8
No such functions
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Copy constructor, assignment operator
So we can NOT do what?
In-class Exercises #1
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Write a recursive member function height() for the BST class
template to return the height of the BST
int height();
// See how concise your implementation could be?
In-class Exercises #2
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Write a recursive member function leafCount() for the BST class
template to return the number of leaf nodes of the BST
int leafCount();
// See how concise your implementation could be?
Questions?
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What is T(n) of search algorithm in a BST?
Why?
What is T(n) of insert algorithm in a BST?
Other operations?
Problem of Lopsidedness
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The order in which items are inserted
into a BST determines the shape of the
BST
Result in Balanced or Unbalanced trees
Insert O, E, T, C, U, M, P
Insert C, O, M, P, U, T, E
Balanced
Unbalanced
Problem of Lopsidedness
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Trees can be totally lopsided
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Suppose each node has a right child only
Degenerates into a linked list
Processing time
affected by
"shape" of tree