Computer Science II

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Computer Science II

• Trees

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What is a Tree?

• A tree is a finite, non-empty set of nodes,
  Tr ? T1 ? T2 ? ? Tn,with the following
  properties
• A designated node of the set, r, is called the
  root of the tree
• The remaining nodes are partioned inton ?
  subsets, T1, T2, , Tn, each of which is a tree

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What does a Tree look like?
Root
Nodes
T2
Empty trees
T1
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Some terminology

• Degree of a node - the number of subtrees
  associated with that node
• Leaf - a node with degree 0 (no subtrees)
• Child - root of a nodes subtree
• Parent - node that represents the immediate
  predecessor to a subtrees root
• Siblings - nodes that share the same parent
• A path is a non-empty sequence of nodes

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More Terminology

• The depth of a node is the length of the path to
  the node from its root
• The height of a tree is the longest path to a
  node from the root
• The height of a node is the longest path from the
  node to a leaf
• An ancestor is another node, above, with a path
  to the node in question
• A descendent is another node, below, which can be
  reached from the node in question

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Breadth-First Traversal
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Breadth-First Traversal
1
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Breadth-First Traversal
1 2
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Breadth-First Traversal
1 2 3
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Breadth-First Traversal
1 2 3 4
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Breadth-First Traversal
1 2 3 4 5
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Pre-Order Traversal
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Pre-Order Traversal
1
14
Pre-Order Traversal
1 2
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Pre-Order Traversal
1 2 4
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Pre-Order Traversal
1 2 4
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Pre-Order Traversal
1 2 4 5
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Pre-Order Traversal
1 2 4 5
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Pre-Order Traversal
1 2 4 5
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Pre-Order Traversal
1 2 4 5 3
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Pre-Order Traversal
1 2 4 5 3
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Post-Order Traversal
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Post-Order Traversal
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Post-Order Traversal
4
25
Post-Order Traversal
4
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Post-Order Traversal
4 5
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Post-Order Traversal
4 5 2
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Post-Order Traversal
4 5 2
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Post-Order Traversal
4 5 2 3
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Post-Order Traversal
4 5 2 3 1
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N-ary Trees

• Either empty, or ...
• has a root, R, and exactly N distinctN-ary
  trees as subtrees

R
T2
T3
T1
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N-ary Trees

• What is the minimum height for an n-ary tree?
• Given an n-ary tree of height X, what is the
  maximum number of nodes?
• Internal nodes?
• Leaf nodes?

R
T2
T3
T1
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Binary Trees

• A binary tree is an n-ary tree where n2

R
TR
TL
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Expression Trees

• An expression tree is an binary tree each
  non-leaf node represents an operator and each
  leaf an operand

1
-
2
4
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Expression Trees

• How would you use an expression tree to represent
  the following?
• 1 2
• (5 - 3) x 2
• (4 x 5) (6 / 3)
• Expression trees provide a convenient method for
  converting among postfix, infix and prefix
  notations

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M-Way Search Trees

• A finite set of keys (elements)
• Either the set T is empty, T?, or ...
• The set consists of n M-way subtrees, T0 thru
  Tn-1, and n-1 keys, k1 thru kn-1, where 2 ? n ? M

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M-Way Search Trees

• All keys at a node are distinct and in order
• All the keys in subtree Ti-1 are less than ki
• All the keys in subtree Ti are greater than ki

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Binary Search Trees
D

• An M-way search tree where M2

E
B
A
C
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Build a Search Tree

• Task to build the tree to the left by adding the
  keys in the following order
• C, A, B, E, D

C
E
A
B
D
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Build a Search Tree

• Start by adding the first key as a lone root node

C
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Build a Search Tree

• Add A
• Since A add A as the root of a new left subtree

C
A
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Build a Search Tree

• Add B
• Since B subtree, we add B to that subtree
• That is, we recursively apply our algorithm to
  the subtree with A at the root

C
A
B
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Build a Search Tree

• Add E
• Since E C, and there is no right subtree,
  we add E as the root of a new right subtree

C
E
A
B
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Build a Search Tree

• Add D
• Since D C, and there is an existing right
  subtree, we add D to that subtree
• Again, a recursive call!

C
E
A
B
D
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Removing a Key

• What if we wanted to remove the key D from this
  search tree?

C
E
A
B
D
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Removing a Key

• What if we wanted to remove the key E from this
  search tree?
• You have 5 minutes to come up with a plan!

C
E
A
B
D
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Removing a Key

• What if we wanted to remove the key C from this
  search tree?
• You have 5 minutes to come up with a plan!

C
E
A
B
D
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Balancing trees

• Given our current scheme for building a binary
  search tree...
• What would the resulting tree look like if we
  added the following keys?A, B, C, D, E, F, G
• What is the height of this tree?
• Could you rearrange these nodes to create a new
  binary search tree with a smaller height?

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Balancing trees

• When searching for a key, the largest number of
  nodes we need to visit is determined by the
  trees height
• Can you tell us why, then, it might be
  advantageous to continually balance these search
  trees?

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AVL Search Trees

• A binary tree is AVL balanced if...
• Its empty, or...
• The heights of the roots two subtrees differ by
  no more than one, and both are AVL balanced
• Tr,TL,TRhL-hR ? 1

r
1
TL
TR
-1
Adelson-Velskii and Landis
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Balancing a TreeSingle Rotation (LL)

• Starting with an AVL balanced tree, ...

B
1
h2
A
0
BR
h1
h
AR
AL
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Balancing a TreeSingle Rotation (LL)

• Add an element as usual, and then ...

B
2
h3
A
1
BR
h2
h1
AR
AL
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Balancing a TreeSingle Rotation (LL)

• Since the AR is to the left of B,

B
2
L
h3
A
1
BR
h2
L
h1
AR
AL
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Balancing a TreeSingle Rotation (LL)

• Make B the right child of A, and ...

A
h3
B
AL
h2
h1
BR
AR
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Balancing a TreeSingle Rotation (LL)

• AR the left child of B

0
A
h2
B
0
AL
h1
h
AR
BR
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Balancing a TreeDouble Rotation (LR)
2
C

• Add a node as usual,

L
h3
-1
A
CR
R
B
-1
AL
h2
R
BL
BR
h
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Balancing a TreeDouble Rotation (LR)
2
C

• Perform a RR rotation on As subtree, and the ...

h2
0
B
CR
h1
A
BR
h
BL
AL
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Balancing a TreeDouble Rotation (LR)
-1
B

• Perform a LL rotation on Cs tree

h2
A
C
0
1
h1
BL
AL
BR
CR
h
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B-Trees

• An M-way tree where
• All leaves are on the same level
• All internal nodes except the root are
  constrained to have at most M non-empty children
  and at least M/2 non-empty children
• The root has at most M non-empty children

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B-Trees

• When to use B-Trees
• Dataset is too large for memory
• Why to use B-Trees
• Reduces disk access
• Nodes (records) are fewer but larger, thus
  reducing number of disk accesses

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B-TreesInsertion

• Insert new element into leaf
• curNode ? leaf
• while (curNode overflow)
• Split curNode into two nodes, same level
• Promote median element up to parent
• curNode ? Parent of curNode

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B-TreesDeletion

• If (element not a leaf) then
• Swap with successor
• curNode ? leaf
• While (curNode underflow)
• Try to redistribute elements from sibling via
  parent
• If possible, merge curNode with sibling and
  element from parent
• curNode ? Parent of curNode