Advanced Trees Part II

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Advanced TreesPart II

• Briana B. Morrison
• Adapted from Alan Eugenio
• William J. Collins

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Topics

• AVL Trees
• Part III
• 2-3-4 Search Trees
• Red Black Trees

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AVL Tree Definition

• AVL trees are balanced.
• An AVL Tree is a binary search tree such that for
  every internal node v of T, the heights of the
  children of v can differ by at most 1.

An example of an AVL tree where the heights are
shown next to the nodes
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Balance Factor

• The balance factor is defined as
• The height of the left subtree minus the height
  of the right subtree
• The tree is balanced if the absolute value of the
  balance factor is lt1
• ht(TL) ht(TR) lt 1

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20 HEIGHT OF LEFT SUBTREE ? HEIGHT OF
RIGHT SUBTREE ?
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NOPE! THE HEIGHT OF THE LEFT SUBTREE IS 1 AND
THE HEIGHT OF THE RIGHT SUBTREE IS 1.
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NOPE THE LEFT AND RIGHT SUBTREES ARE NOT AVL
TREES, SO THE TREE IS NOT AN AVL TREE.
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NOPE! THE HEIGHT OF THE LEFT SUBTREE IS 2 AND
THE HEIGHT OF THE RIGHT SUBTREE IS 0.
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Height of an AVL Tree

• Fact The height of an AVL tree storing n keys is
  O(log n).
• Proof Let us bound n(h) the minimum number of
  internal nodes of an AVL tree of height h.
• We easily see that n(1) 1 and n(2) 2
• For n gt 2, an AVL tree of height h contains the
  root node, one AVL subtree of height n-1 and
  another of height n-2.
• That is, n(h) 1 n(h-1) n(h-2)
• Knowing n(h-1) gt n(h-2), we get n(h) gt 2n(h-2).
  So
• n(h) gt 2n(h-2), n(h) gt 4n(h-4), n(h) gt 8n(n-6),
  (by induction),
• n(h) gt 2in(h-2i)
• Solving the base case we get n(h) gt 2 h/2-1
• Taking logarithms h lt 2log n(h) 2
• Thus the height of an AVL tree is O(log n)

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• ITEMS NOT IN THE SUBTREE OF THE ITEM ROTATED
  ABOUT ARE UNAFFECTED BY THE ROTATION.
• A ROTATION TAKES CONSTANT TIME.
• BEFORE AND AFTER A ROTATION, THE TREE IS STILL A
  BINARY SEARCH TREE.
• THE CODE FOR A LEFT ROTATION IS SYMMETRIC TO THE
  CODE FOR A RIGHT ROTATION SIMPLY SWAP left AND
  right.

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• Now lets look at some examples of inserting into
  an AVL tree to get an idea of how all this
  rotation stuff works

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Example Insert numbers 1-7 into AVL
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Now add 16 and 15
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Now an Example with State Abbr.
Insert DE
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Insert OH
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Insert MA
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Insert IL and MI
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Insert IN
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Insert NY
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Insert VT
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Insertion in an AVL Tree

• Insertion is as in a binary search tree
• Always done by expanding an external node.
• Example

cz
ay
bx
w
before insertion
after insertion
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Restructuring (as Single Rotations)

• Single Rotations

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Restructuring (as Double Rotations)

• double rotations

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Removal in an AVL Tree

• Removal begins as in a binary search tree, which
  means the node removed will become an empty
  external node. Its parent, w, may cause an
  imbalance.
• Example

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17
62
78
50
88
48
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before deletion of 32
after deletion
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Rebalancing after a Removal

• Let z be the first unbalanced node encountered
  while travelling up the tree from w. Also, let y
  be the child of z with the larger height, and let
  x be the child of y with the larger height.
• We perform restructure(x) to restore balance at
  z.
• As this restructuring may upset the balance of
  another node higher in the tree, we must continue
  checking for balance until the root of T is
  reached

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44
az
44
78
17
62
w
by
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50
88
78
50
cx
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88
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Analysis of AVL Trees

• The number of recursive calls to insert a new
  node can be as large as the height of the tree.
• At most one (single or double) rotation will be
  done per insertion.
• A rotation improves the balance of the tree, so
  later insertions are less likely to require
  rotations.
• It is very difficult to find the height of the
  average AVL tree, but the worst case is much
  easier. The worst-case behavior of AVL trees is
  essentially no worse than the behavior of random
  search trees.

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• Empirical evidence suggests that the average
  behavior of AVL trees is much better than that of
  random trees, almost as good as that which could
  be obtained from a perfectly balanced tree.
• Algorithms for manipulating AVL trees are
  guaranteed to take no more than about 44 percent
  more time than the optimum. In practice, AVL
  trees do much better than this on average,
  perhaps as small as lg n 0.25.

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Running Times for AVL Trees

• a single restructure is O(1)
• using a linked-structure binary tree
• find is O(log n)
• height of tree is O(log n), no restructures
  needed
• insert is O(log n)
• initial find is O(log n)
• Restructuring up the tree, maintaining heights is
  O(log n)
• remove is O(log n)
• initial find is O(log n)
• Restructuring up the tree, maintaining heights is
  O(log n)