Dr' Wenzhan Song

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Dr. Wenzhan Song Assistant Professor, Computer
Science
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Chapter 4 Trees

• See how trees are used to implement the file
  system of several popular operating systems
• See how trees can be used to evaluate arithmetic
  expressions
• See how to use trees to support searching
  operations in O(log N) worst-case bounds
• Discuss and use the set and map classes

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Preliminaries
Generic tree
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Preliminaries
A tree example
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Preliminaries
First child/next sibling representation of the
tree in previous example
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Preliminaries
UNIX directoty
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Preliminaries
Unix directory with file sizes obtained via
postorder traversal
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Binary Trees
Generic binary tree
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Binary Trees
Worst case binary tree
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Binary Trees
Expression tree for (abc)((def)g)
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Binary Search Trees
Two binary trees (on the left tree is a search
tree)
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Binary Search Trees
binary search trees before and after inserting 5
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Binary Search Trees
Deletion of a node (4) with one child, before and
after
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Binary Search Trees
Deletion of a node (2) with two children, before
and after
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Binary Search Trees
A randomly generated binary search tree
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Binary Search Trees
Binary search tree after insert/remove
pairs
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AVL Trees
A bad binary search tree. Requiring balance at
the root is not enough
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AVL Trees
Two binary search tree. Only the left tree is AVL.
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AVL Trees
Smallest AVL tree of height 9
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AVL Trees
Single rotation to fix case 1
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AVL Trees
AVL property destroyed by insertion of 6, then
fixed by a single rotation
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AVL Trees
Single rotation fixes case 4
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AVL Trees
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AVL Trees
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AVL Trees
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AVL Trees
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AVL Trees
Single rotation fails to fix case 2
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AVL Trees
Left-right double rotation to fix case 2
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AVL Trees
Right-left double rotation to fix case 3
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AVL Trees
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AVL Trees
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AVL Trees
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AVL Trees
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AVL Trees
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AVL Trees

• Summarized Tips
• Case 1 (left-left) or Case 4 (right-right)
• Step 1 Find the deepest unbalanced subtree root,
  say node x
• Step 2 Find the child of node x, say node y,
  whos subtree is taller
• Step 3 Rotate subtree with node x promote node
  y to replace the position of node x, then rotate
  naturally only one choice!
• Case 2 (left-right) or Case 3 (right-left)
• Step 1 Find the deepest unbalanced subtree root,
  say node x
• Step 2 Find the child of node x, say node y,
  whos subtree is taller
• Step 3 Find the child of node y, say node z,
  whos subtree is taller
• Step 4 Rotate subtree with node y promote node
  z to replace the position of node y, then rotate
  naturally only one choice!
• Step 5 Rotate subtree with node x promote node
  z to replace the position of node x, then rotate
  naturally only one choice!
• (Alternative of Step 4 and 5 promote node z to
  replace node x, than attach x and y to left and
  right of node z, and other subtrees A,B,C,D are
  attached naturally only one choice)

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Splay Trees
Figure 4.47 Zig-zag
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Splay Trees
Figure 4.48 Zig-zig
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Splay Trees
The first splay step is at k1, and is clearly a
zig-zag
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Splay Trees
The next splay step at k1 is a zig-zig
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Splay Trees
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Splay Trees
Result of splaying at node1
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Splay Trees
Result of splaying at node 1 a tree of all left
children
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Splay Trees
Result of splaying the previous tree at node 2
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Splay Trees
Result of splaying the previous tree at node 3
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Splay Trees
Result of splaying the previous tree at node 4
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Splay Trees
Result of splaying the previous tree at node 5
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Splay Trees
Result of splaying the previous tree at node 6
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Splay Trees
Result of splaying the previous tree at node 7
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Splay Trees
Result of splaying the previous tree at node 8
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Splay Trees
Result of splaying the previous tree at node 9
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Splay Trees

• Add operation does not involve splaying
• Delete operation does
• (1) splay the to-be-delete node to the root,
  then delete the root and result TL and TR.
• (2) Rotate the largest element of TL to the
  root, then TL will have a root with no right
  child.
• (3) Let TR be the right child of new root

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B-Trees
5-ary tree of 31 nodes has only three levels
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B-Trees
B-tree of order 5
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B-Trees
B-tree after insertion of 57 into the tree in the
previous fig
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B-Trees
Insertion of 55 into the B-tree in previous Fig
causes a split into two leaves
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B-Trees
Insertion of 40 into the B-tree in previous Fig
causes a split into two leaves and then a split
of the parent node
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B-Trees
B-tree after the deletion of 99 from the B-tree
in previous Fig
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Red-Black Trees

• It is a binary search tree with the following
  coloring properties
• Every node is colored either red or black
• The root is black
• If a node is red, its children must be black
• Every path from a node to a NULL pointer must
  contain the same number of black nodes
• Hence the height is at most 2log(N1)

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Red-Black Trees
Red node
Fig. 12.9 Example of a red black tree (insertion
sequence 10, 85, 15, 70, 20, 60, 30, 50, 65, 80,
90, 40, 5, 55)
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Bottom-up Insertion
Fig. 12.10 Zig rotation and zig-zag rotation work
if S is black
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Top-down Insertion
Fig. 12.10 Color flip only if Xs parent is red
do we continue with a rotation Then apply the
rotation in previous slides
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Top-down Insertion
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Top-down Deletion
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Trees
Tree for Exercises 4.1 to 4.3
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Trees
Tree for Exercise 4.8
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Trees
Tree for Exercise 4.27
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Trees
Tree for Exercise 4.43
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Trees
Two isomorphic trees