Binary search trees

1
Binary search trees

• Definition
• Binary search trees and dynamic set operations
• Expected height of BST
• Balanced binary search trees
• Tree rotations
• Red-black trees

2
Binary search trees

• Basic tree property
• For any node x
• left subtree has nodes x
• right subtree has nodes x

3
BSTs and Dynamic Sets

• Dynamic set operations and binary search trees
• Search(S,k)
• Insert(S,x)
• Delete(S,x)
• Minimum or Maximum(S)
• Successor or Predecessor (S,x)
• List All(S)
• Merge(S1,S2)

4
Dynamic Set Operations

• Listall(T)
• time to list?
• Search(T,k)?
• search time?
• Minimum(T)? Maximum(T)?
• search time?
• Successor(T,x)? Predecessor(T,x)?
• Search time
• Simple Insertion(T,x)

5
Simple deletion

• Delete(T,x) Three possible cases
• a) x is a leaf
• b) x has one child
• c) x has two children Replace x with
  successor(x).
• Successor(x) has at most one child (why?)
• Use step a or b on successor(x)

6
Simple binary search trees

• What is the expected height of a binary search
  tree?
• Difficult to compute if we allow both insertions
  and deletions
• With insertions, analysis of section 12.4 shows
  that expected height is O(log n)
• Implications about BSTs as dynamic sets?

7
Definitions and Assumptions

• We assume that the n items are distinct 1-n
• We assume no deletions
• We assume that each of the n! permutations of the
  n items are equally likely
• Let Xi be the height of a tree with i items
• Let
• Let Zi be an indicator variable that the first
  item inserted is item i

8
Recurrence relation for Xn and Yn

• Xn Si Zi (1 max(Xi-1, Xn-i))
• Justify this
• Yn Si Zi (2 max(Yi-1,Yn-i))
• Analysis on handout to get

9
Tree-Balancing Algorithms

• Tree rotations
• Red-Black Trees
• Splay Trees
• Others
• AVL Trees
• 2-3 Trees and 2-3-4 Trees

10
Tree Rotations
Right Rotate(T,A)
B
A
T1
A
B
T3
T2
T3
T1
T2
Left Rotate(T,B)
11
Red-Black Trees

• All nodes in the tree are either red or black.
• Root node is black.
• Every leaf (null-child) is included and black.
• All red nodes must have two black children.
• Every path from any node x (including the root)
  to a leaf must have the same number of black
  nodes.
• How balanced of a tree will this produce?
• How hard will it be to maintain?

12
Example Red-Black Tree
13
Insertion(T,z)

• Find place to insert using simple insertion
• Color node z as red
• Fix up tree so that it is still a red-black tree
• What needs to be fixed?
• Problem 1 z is root (first node inserted)
• Minor detail
• Problem 2 parent(z) is red

14
RB-Insert-Fixup

• Situation parent(z) is red and z is red
• Case 1 uncle(z) is red
• Then make both uncle(z) and parent(z) black and
  p(p(z)) red and recurse up tree

15
RB-Insert-Fixup(parent(z) is right child of
parent(parent(z)))

• Situation parent(z) is red and z is red
• Case 2 uncle(z) is black and z is a left child
• Right rotate to make into case 3

16
RB-Insert-Fixup(parent(z) is right child of
parent(parent(z)))

• Situation parent(z) is red and z is red
• Case 3 uncle(z) is black and z is a right child
• Left rotate to make B root of tree

17
RB-Insert-Fixup Analysis(parent(z) is right
child of parent(parent(z)))

• Situation parent(z) is red and z is red
• Case 1 no rotations, always moving up tree
• Cases 2 and 3 At most 2 rotations total and tree
  ends up balanced
• No more need to fix up once these cases are met
• Total cost at most 2 rotations and log n
  operations

18
Delete(T,z)

• Find node y to delete using simple deletion
• Let x be a child of y if such a child exists
  (otherwise x is a null child)
• If y is black, fix up tree so that it is still a
  red-black tree
• What needs to be fixed?
• Problem 1 y was root, so now we might have red
  root
• Problem 2 x and parent(y) are both red
• Problem 3 Removal of y violates black height
  properties of paths that used to go through y