Post on 20-Dec-2015
CSE 326: Data StructuresBinary Search Trees
Ben Lerner
Summer 2007
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Tree CalculationsRecall: height is max
number of edges from root to a leaf
Find the height of the tree...
t
Can you do this in less than O(n) time?
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Tree Calculations Example
A
E
B
D F
C
G
IH
KJ L
M
L
N
How high is this tree?
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Binary Trees• Binary tree is
– a root– left subtree (maybe
empty) – right subtree (maybe
empty)
• Representation:
A
B
D E
C
F
HG
JI
Data
right pointer
leftpointer
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Binary Tree: RepresentationA
right pointer
leftpointer A
B
D E
C
F
Bright
pointerleft
pointer
Cright
pointerleft
pointer
Dright
pointerleft
pointer
Eright
pointerleft
pointer
Fright
pointerleft
pointer
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Binary Tree: Special Cases
A
B
D E
C
GF
IH
A
B
D E
C
F
A
B
D E
C
GF
Full Tree
Complete Tree Perfect Tree
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Binary Tree: Some Numbers!For binary tree of height h:
– max # of leaves:
– max # of nodes:
– min # of leaves:
– min # of nodes:
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More Recursive Tree Calculations:
Tree TraversalsA traversal is an order for
visiting all the nodes of a tree
Three types:• Pre-order: Root, left subtree, right
subtree
• In-order: Left subtree, root, right subtree
• Post-order: Left subtree, right subtree, root
+
*
2 4
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(an expression tree)
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Traversals
void traverse(BNode t){
if (t != NULL)
traverse (t.left);
print t.element;
traverse (t.right);
}
} Which kind of traversal is this?
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ADTs Seen So Far
• Stack– Push– Pop
• Queue– Enqueue– Dequeue
What about decreaseKey?
• Priority Queue– Insert– DeleteMin
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The Dictionary ADT
• Data:– a set of
(key, value) pairs
• Operations:– Insert (key,
value)– Find (key)– Remove (key) The Dictionary ADT is sometimes
called the “Map ADT”
• blernerBen Lerner,OH: Tues, 10:45—12:00CSE 212
• ioannisIoannis Giotis,OH: M 11:00CSE 430
• gyanitGyanit Singh,OH: M 11:00CSE 430
insert(blerner, ….)
find(ioannis)• ioannis Ioannis Giotis, …
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A Modest Few Uses
• Sets
• Dictionaries
• Networks : Router tables
• Operating systems : Page tables
• Compilers : Symbol tables
Probably the most widely used ADT!
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Implementations
• Unsorted Linked-list
• Unsorted array
• Sorted array
insert deletefind
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Binary Search Tree Data Structure
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121062
115
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13
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• Structural property– each node has 2 children– result:
• storage is small• operations are simple• average depth is small
• Order property– all keys in left subtree smaller
than root’s key– all keys in right subtree larger
than root’s key– result: easy to find any given key
• What must I know about what I store?
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Example and Counter-Example
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21BINARY SEARCH TREE NOT ABINARY SEARCH TREE
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Find in BST, RecursiveNode Find(Object key, Node root) { if (root == NULL) return NULL;
if (key < root.key) return Find(key, root.left); else if (key > root.key) return Find(key, root.right); else return root;}
2092
155
10
307 17
Runtime:
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Find in BST, IterativeNode Find(Object key, Node root) {
while (root != NULL && root.key != key)
{ if (key < root.key) root = root.left; else root = root.right; }
return root;}
2092
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307 17
Runtime:
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Insert in BST
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307 17
Runtime:
Insert(13)Insert(8)Insert(31)
Insertions happen only at the leaves – easy!
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BuildTree for BST• Suppose keys 1, 2, 3, 4, 5, 6, 7, 8, 9 are inserted
into an initially empty BST. Runtime depends on the order!
– in given order
– in reverse order
– median first, then left median, right median, etc.
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Bonus: FindMin/FindMax
• Find minimum
• Find maximum2092
155
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307 17
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Deletion in BST
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155
10
307 17
Why might deletion be harder than insertion?
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Delete Operation
• Delete is a bit trickier…Why?• Suppose you want to delete 10• Strategy:
– Find 10– Delete the node containing 10
• Problem: When you delete a node,what do you replace it by?
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Delete Operation
• Problem: When you delete a node,what do you replace it by?
• Solution:– If it has no children, by NULL– If it has 1 child, by that child– If it has 2 children, by the node with
the smallest value in its right subtree(the successor of the node)
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5 24
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Delete “5” - No children
Find 5 node
Then Freethe 5 node and NULL the pointer to it
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Delete “24” - One child
Find 24 node
Then Freethe 24 node and replace the pointer to it witha pointer to itschild
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Delete “10” - two children
Find 10,Copy the smallestvalue inright subtreeinto the node
Then recursivelyDelete node with smallest valuein right subtreeNote: it does nothave two children
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5 24
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Delete “11” - One child
Remember11 node
Then Freethe 11 node and replace the pointer to it witha pointer to itschild
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11 97
5 24
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5 24
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Runtimes
• Find? Insert? Delete?
• What is the average height of a BST?
• What is the maximum height?
• What happens when we insert nodes in sorted order?
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Balanced BST
Observation• BST: the shallower the better!• For a BST with n nodes
– Average height is O(log n)– Worst case height is O(n)
• Simple cases such as insert(1, 2, 3, ..., n)lead to the worst case scenario
Solution: Require a Balance Condition that1. ensures depth is O(log n) – strong enough!2. is easy to maintain – not too strong!
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Potential Balance Conditions1. Left and right subtrees of the root
have equal number of nodes
2. Left and right subtrees of the roothave equal height
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Potential Balance Conditions3. Left and right subtrees of every node
have equal number of nodes
4. Left and right subtrees of every nodehave equal height
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The AVL Balance ConditionLeft and right subtrees of every nodehave equal heights differing by at most 1
Define: balance(x) = height(x.left) – height(x.right)
AVL property: –1 balance(x) 1, for every node x
• Ensures small depth– Will prove this by showing that an AVL tree of height
h must have a lot of (i.e. O(2h)) nodes• Easy to maintain
– Using single and double rotations
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The AVL Tree Data Structure
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Structural properties
1. Binary tree property
2. Balance property:balance of every node isbetween -1 and 1
Result:
Worst case depth isO(log n)
Ordering property– Same as for BST
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