The outcome of any serious research can only be to make two questions grow where only one grew befor

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• The outcome of any serious research can only be
  to make two questions grow where only one grew
  before.
• Thorstein Veblen (1857 - 1929)

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CSE 502NFundamentals of Computer Science

• Fall 2004
• Lecture 10
• Binary Search Trees (CLRS 12)
• Introduction to Red-Black Trees (CLRS 13)

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Binary Search Trees

• A binary search tree is a data structure that
  supports the dynamic set operations Insert,
  Search, Delete, Minimum, Maximum, Predecessor,
  and Successor
• Used both as a dictionary and a priority queue
• Binary-search-tree property let x be a node in a
  BST if y is a node in the left subtree of x,
  then keyy keyx if y is a node in the right
  subtree of x, then keyx keyy

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Tree walking

• Inorder tree walk is a recursive algorithm that
  prints the keys stored in the BST in sorted order
• Prints the root of a subtree after visiting the
  left subtree and prior to visiting the right
  subtree
• T(n) Q(n)
• Preorder tree walk prints the subtree root prior
  to visiting the left and right subtrees
• Postorder tree walk prints the subtree root after
  visiting the left and right subtrees

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BST Queries

• Given a pointer to the root node and a key k, a
  BST query returns a pointer to the node
  containing the key k or NIL if the BST does not
  contain the key
• Tree-Search is a recursive query routine
• T(n) O(h) where h is the height of the BST
• Iterative-Tree-Search unrolls the recursion
  using a while loop
• More efficient on most computers why?
• T(n) O(h) where h is the height of the BST

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Minimum Maximum

• The leftmost node in the tree is the minimum
• The rightmost node in the tree is the maximum
• Both properties guaranteed by the binary search
  tree property
• Both run in O(h) where h is the height of the BST

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Successor Predecessor

• Successor of x is the next node in the sorted
  order
• Returns next node printed in an Inorder Tree Walk
• BST structure allows successor to be found
  without comparing keys
• T(n) O(h) where h is the height of the BST
• Predecessor is a symmetric operation

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Insertion

• Tree-Insert must maintain binary search tree
  property
• T(n) O(h) where h is the height of the BST

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Deletion

• Tree-Delete must maintain binary search tree
  property
• Considers three cases
• z has no children
• Modify parents child pointer
• z has one child
• Modify childs parent pointer
• Modify parents child pointer
• z has two children
• Replace z with its successor
• Modify pointers
• T(n) O(h) where h is the height of the BST

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Running Time

• Search, Minimum, Maximum, Successor, Predecessor,
  Insert, and Delete run in O(h) time where h is
  the height of the BST
• Worst-case can be Q(n) for linear chain of nodes
• Tree height is primary determinant of running
  time
• What determines tree height?
• Can we make any claims about the expected height
  of a tree containing n keys?
• Randomly built binary search trees have expected
  height O(lg n)
• Can we bound the worst-case tree height to O(lg
  n)?

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Red-Black Trees

• Red-black trees are one of many balanced search
  tree data structures
• For n nodes a red-black tree has height at most
  2(lg n 1) O(lg n)
• Assigns each node a color (red or black)
• Red-black properties
• Every node is either red or black
• The root node is black
• Every leaf (NIL) is black
• If a node is red, then both its children are
  black
• For each node, all paths from the node to
  descendant leaves contain the same number of
  black nodes
• Insert and delete operations must be modified in
  order to maintain red-black properties (maximum
  height, balance condition)
• Use rotations to modify structure of tree while
  maintaining binary search tree property

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Rotations
Left-Rotate(T,x)
x
y
g
y
a
x
Right-Rotate(T,y)
b
g
a
b