Trees

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Trees

• Trees
• Binary tree and Binary tree traversals
• Expression trees
• Infix, Postfix, Prefix notation
• Binary Search Trees
• Operations supported
• Complexity

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(Directed) Tree
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Definition of Trees

• /\ is a tree. It is called an empty tree.
• If is a node and
• are non-empty trees where k ? 1, then
• is a tree.

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Notation about Trees

• root, subtrees
• parent, child, sibling
• leaves, internal nodes
• edge
• path, length
• ancestor, descendant
• depth, height
• Root at depth 0,
• One node tree height 0

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Tree criteria

• Always a single root
• A single path to every node
• Doesnt matter how we draw it

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Trees ?
4
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Game tree

• 2 7
• 1 3
• 8 6 X

Start state for the 8-puzzle X marks the empty
square.
Can move the 3 tile into the empty square, or,
can move the 6 tile into it.

• 2 7
• 1 X
• 8 6 3

• 2 7
• 1 3
• 8 X 6

Goal of the game is to place all tiles in some
special order.
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Binary Trees

• Binary tree every node has at most 2 children
  (left child and right child)
• N-ary tree no node has more than N children.
• Advantages?
• Disadvantages?
• Equivalencies?

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Some Properties of Binary Trees

• Suppose a binary tree has n nodes, and height h.
• The total numbers of nodes h1 ?n ?2h1-1
• The height of the tree (from inequality above)
• log(n1)-1 ? h ? (n-1)

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Implementation of Binary Trees
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Binary Trees
node
LL left link RL Right Link
T
(((ab)(c(d-3)))
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Implementation of N-ary Trees
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Implementation of Trees
Right link to siblings left link to successors
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Some applications

• Expressions ( (731218) (4/3) )
• HTML or XML
• Word documents

3
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Traversing a tree

• Traversing a tree is to visit every node in the
  tree

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Binary Tree Traversal

• Preorder
• Root
• Left
• Right
• In order
• Left
• Root
• Right
• Postorder
• Left
• Right
• Root

Preorder (T) if TNIL then
return() Print(T-INFO) Preorder
(T-LL) Preorder (T-RL) Inorder (T) if
TNIL then return() Inorder (T-LL) Print
(T-INFO) Inorder (T-RL) Postorder (T)
if TNIL then return() Postorder
(T-LL) Postorder (T-RL) Print (T-INFO)
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Preorder Traversal
Preorder (T) if TNIL then
return() Print(T-INFO) Preorder
(T-LL) Preorder (T-RL)
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Used for copying a tree

• As in copy constructor for tree class
• Copy the root
• Recursively copy left subtree
• Recursively copy right subtree
• Always have parent node to link new nodes to

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Inorder Traversal
Inorder (T) if TNIL then return() Inorder
(T-LL) Print (T-INFO) Inorder (T-RL)
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Postorder Traversal
( )
postorder
Postorder (T) if TNIL then
return() Postorder (T-LL) Postorder
(T-RL) Print (T-INFO)
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Used to destruct tree

• Return leaf nodes to heap via delete in a bottom
  up fashion
• When children are returned to heap, then return
  the parent to heap
• More difficult to do in preorder

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Example of Traversals
Preorder a b c - d 3 prefix
notation Postorder a b c d 3 - Postfix
notation Inorder a b c d - 3 Infix
notation Prefix and postfix notation
unambiguous Infix notation ambiguous.
parenthesis necessary
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Scene graph from graphics

• Tree (graph) nodes are visited in preorder (top
  down)
• Node might set light or place the camera
• Node might set properties, such as color and
  shine
• Node might define rotation or translation
• Node might define object geometry to render

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Expression Trees
63

• Postfix
• 1 2 3 4 5
• Prefix
• 1 2 3 4 5
• Infix
• 1 2 3 4 5 30
• ( 1 ( 2 3 ) ) ( 4 5 ) 63

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9


6
1

4
5
2
3
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Postfix to Expression Tree

• Similar to evaluating postfix expressions
• Evaluating a postfix expression
• Scan the input one symbol at a time
• If it is a number
• push it onto the stack
• If it is an operator,
• pop 2 numbers from the stack
• apply the operator to the 2 numbers
• push the result back onto the stack

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Constructing an expression tree

• 1 2 3 4 5 -
• 1 2 3 4 5 1
• 1 2 3 4 5 1 2
• 1 2 3 4 5 1 2 3
• 1 2 3 4 5 1 6

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Binary Tree Reconstruction

• For a given preorder sequence of binary tree, is
  the inorder sequence unique?
• Given a preorder sequence of binary tree, we can
  construct a lot of BTs according to the
  corresponding inorder sequence.

2
1
3
2
1
3
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Example

• 1, 2, 3
• Preorder sequence 123
• Possible inorder sequences are
• 123, 132, 213, 231, 321

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Binary search trees

• Store keys
• all keys in left subtree keys in right subtree

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Search Tree
LEFT sorted
list




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Example
How do we find 8 in this tree?
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Search Operations

• Find( tree, key)
• the pointer to the key (or null pointer)
• FindMin( tree), FindMax( tree)
• What is the running time?

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Find, FindMin, FindMax

• Find
• Find Min
• Find Max

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How to insert?

• What if I insert
• 14
• 99

Insert(X, var T) if(T nil) T
newnode(X) else if X
Insert(X, T.LL) if X T.item
Insert(X, T.RL)
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Delete

• 3 cases
• What are they?
• Which are easy?

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Delete 2 children
2
Deletemin from Right child
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Deletion
delete(var T,x) if T NIL then return x
does not exists in T else if (x INFO )
then delete (T-LL,x) else if (x T-INFO )
then delete (T-RL,x) else // T-INFO
x if (T - LL NIL) then T T- RL else
if (T - RL NIL) then T T- LL else
// T has left child, // max of left
child becomes the root
y deletemin (T-RL) T- INFO y
Q How to write delete with
deletemin?
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Deletemin
int deletemin(var T) if T NIL then
return tree is already empty else if
(T - LL NIL) // T has only right child so T
is the min x T - INFO
T T - RL else //
T has a left child, follow left child x
deletemin (T-LL) return
(x) Question How to do deletemax?
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Complexity

• search, insertion, deletemax, deletion
• average case O(log n) ---
• But, only if random insertion!!!!
• worst case O(n)

Question How to improve the performance?
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Balancing Search Tree

• AVL Tree
• for every node in the tree, the height of the
  left and right subtree can differ by at most 1.
• How to keep it balanced?
• Rotation (single, or double)
• average depth and worst case depth are O(logn)
• Splay Tree
• life time total complexity for m operations is
  O(mlogn), where n is the size of the tree
• simpler to implement than AVL tree
• B Tree
• each node has between 2 and M children
• Design for small max path
• good for external storage (disks etc.) data
  structures

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Testing balanced

• boolean is_balanced (T, var height)
• if TNIL then height -1 return true
• else
• left_b is_balanced (T-LL,
  left_height)
• right_b is_balanced (T-RL,
  right_height)
• height max (left_height,
  right_height) 1
• if ( left_b and right_b and
  (abs(left_height -
  right_height)
• then return(true)
• else return (false)

What is the criteria?
2
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Other things
int height (T) if TNIL then return
(-1) else return ( max (height(T-LL),
height(T-RL)) 1)

• int count(T)
• if TNIL then return (0)
• else return (count(T-LL) count(T-RL) 1)

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Good Trees and Bad Trees
h ?(log n)
h ?(n)