Chapter 12 Binary Search Trees

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Chapter 12 Binary Search Trees
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Introduction

• Search trees support SEARCH, MINIMUM, MAXIMUM,
  PREDECESSOR, SUCCESSOR, INSERT, AND DELETE
• Dictionary and priority queue
• Basic operations take time proportional to the
  height of the tree
• Complete binary tree ?(lg n)
• Linear-chain tree ?(n)
• Expected height of a randomly built binary search
  tree O(lg n)

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What Is A Binary Search Tree?
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Definition

• Binary search tree (BST)
• Binary tree
• Represented by a linked data structure ? each
  node is an object
• Key satellite data
• Pointers left ? left child, right ? right child,
  p ? parent
• 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
  keyy ? keyx

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BST Example
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Tree Traversal

• Inorder tree walk
• The key of the root of a subtree is printed
  between the values in its left subtree and those
  in its right subtree
• Can print out all the keys in a BST in sorted
  order
• (a) 2, 3, 5, 5, 7, 8 (b) 2, 3, 5, 5, 7, 8
• Preorder tree walk
• The key of the root of a subtree is printed
  before the values in its left subtree and those
  in its right subtree
• (a) 5, 3, 2, 5, 7, 8 (b) 2, 3, 7, 5, 5, 8
• Postorder tree walk
• The key of the root of a subtree is printed after
  the values in its left subtree and those in its
  right subtree
• (a) 2, 5, 3, 8, 7, 5 (b) 5, 5, 8, 7, 3, 2

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InOrder-Tree-Walk
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Computation Time of INORDER-TREE-WALK(x)

• Theorem 12.1. If x is the root of an n-node
  subtree, then the call INORDER-TREE-WALK(x) takes
  ?(n) time
• Use the substitution method
• T(n) T(k) T(n-k-1) d T(0)c
• Show that T(n) ?(n) by proving that T(n)
  (cd)nc
• For n0 ? (cd) 0 c c
• For n gt 0
• T(n) T(k) T(n-k-1) d ((cd)kc)((cd)(n
  -k-1)c) d (cd)nc-(cd)cd (cd)n c

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Query A Binary Search Tree
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Searching
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Searching Example
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Minimum and Maximum

• The element with the minimum/maximum key can be
  found by following left/right child pointers from
  the root until NIL is encountered

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Successor And Predecessor

• Assume distinct keys
• The successor of a node x is the node with the
  smallest key greater than keyx
• Right subtree nonempty ? successor is the
  leftmost node in the right subtree (Line 2)
• Otherwise, the successor is the lowest ancestor
  of x whose left child is also an ancestor of x ?
  go up the tree until we encounter a node that is
  the left child of its parent (Lines 3-7)

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TREE-SUCCESSOR(X)
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Insertion and Deletion
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Insertion
Should be inserted in the left subtree
Should be inserted in the right subtree
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Insertion Example
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Insert an item with key 13into a BST
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5
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19
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9
2
15
17
13
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Deletion

• Consider three cases for the node z to be deleted
• z has no children modify pz to replace z with
  NIL as its child
• z has only one child splice out z by making a
  new link between its child and its parent
• z has two children splice zs successor y, which
  has no left child and replace zs key and
  satellite data with ys key and satellite

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TREE-DELETE Algorithm

• TREE-DELETE Algorithm
• Lines 1-3 determines a node y to splice out (z
  itself or zs successor)
• Lines 4-6 x is set to the non-NIL child of y, or
  to NIL if y has no children
• Lines 7-13 splice out y by modifying pointers in
  py and x
• Special care for when xNIL or when y is the root
• Lines 14-16 if the successor of z was the node
  spliced out, ys key and satellite data are moved
  to z, overwriting the previous key and satellite
  data
• Line 17 return node y for recycle it via the
  free list

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Randomly Built BST

• All the basic operations on a BST run in O(h)
  time
• If the elements are inserted in strictly
  increasing order height n-1
• Exercise B.5-4 h ? ?lg n?
• The average case is much closer to the best case
• Complicated if both insertion and deletion are
  used to create a BST
• Randomly Built BST on n distinct keys
• A BST arises from inserting the keys in random
  order into an initially empty tree, where each of
  the n! permutations of the input keys is equally
  likely
• Expected height of a randomly built BST on n
  distinct keys is O(lg n)
• Section 12.4 (Self-Study)