AVL Trees II
Implementation
1
AVL Tree ADT
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A binary search tree in which the balance
factor of each node is 0, 1, of -1.
Basic Operations
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Construction, empty, search, and traverse
are the same as for a BST.
Insert a new item in such a way that the
height-balanced property is maintained.
Delete a item in such a way that the heightbalanced property is maintained.
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Left for advanced course!
Discussed in Brozdek.
2
AVL Trees
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Class to represent AVL tree nodes
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Includes new data member for the balance factor
3
What do we need to know?
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When we look at a tree on paper, it is
fairly easy to determine if the tree has
become unbalanced.
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Also what to do to fix the problem.
When implementing the ADT, we have to
work with local information.
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What do we have available as we descend
the tree to add a node and return?
What do we need to know at each node?
4
Observations
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When we add a node to an AVL tree, the parent node
will either be a leaf or have a single child.
If the parent node is a leaf, its balance factor prior to
the addition is 0.
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After the addition, the balance factor absolute value will be 1.
Ancestor nodes’ balance factor may increase, decrease, or
remain the same.
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Rebalancing may or may not be necessary.
If the parent node has a single child, its absolute
balance factor prior to the addition will be 1.
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After the addition, it will be 0.
Ancestor nodes’ balance factors will not change.
Rebalancing will not be necessary.
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Observations
Given that the tree was an admissible AVL tree before
we added a node:
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Adding a node always changes the balance factor of
the parent of the new node.
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Absolute value 0 to 1 or 1 to 0.
The parent node will never become unbalanced due
to an addition.
The grandparent is the first node (going up the tree)
that can become unbalanced due to an addition.
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An ancestor further up the tree could be the first node to
become unbalanced.
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See addition of MA in previous presentation.
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When does a node become unbalanced?
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A node will become unbalanced due to an
addition if the following conditions are true:
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It already had an balance factor of +1 or -1.
In searching for the point to do the addition we
descended into its subtree that already had the
greater height.
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Left subtree if balance factor was +1.
Right subtree is balance factor was -1.
Adding the node increased the height of that
subtree.
We need to know if adding a node to a subtree
increased the height of the subtree.
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Did the subtree height increase?
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We need to know if adding a node to a subtree
increased the height of the subtree.
The (recursive) insert function must report
back to its caller whether adding a new node
increased the height of the subtree to which it
was added.
At each step we can then determine if the node
has become unbalanced.
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Whether or not we need to do a rotation.
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When does a node become unbalanced?
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Adding a node increased the height of the
subtree at a node along the path from the new
node back to the root if:
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The addition increased the height of the subtree
rooted at the parent of the new node.
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The addition increased the height of the subtree
rooted at each intermediate node.
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We added the node to a leaf.
Balance factor at each intermediate node was 0
If a node previously had a nonzero balance factor and
adding the new node increased the height of its higher
subtree, the addition made that node unbalanced.
9
Rebalancing the Tree
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If adding a node increased the height of a
subtree and any node became unbalanced
because of the increase, we will do a rotation
at the nearest ancestor that became
unbalanced.
The rotation reduces the height of the subtree
rooted at that position.
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Will ensure that the subtree at that position does
not increase in height.
No further rotations will be needed.
10
Rebalancing the Tree
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We will use a recursive search to find the
node at which to attach the new node.
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Previously we used a loop.
When we find the place to attach the
new node, after attaching it, return
values saying
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Whether or not the subtree height increased.
On which side the new node was added.
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Required in order to determine if a double
rotation is needed.
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Rebalancing the Tree
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At each step in the sequence of recursive
calls, report back the same information:
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Did the subtree rooted at this position
increase in height?
Which direction did we descend to do the
addition?
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Rebalancing the Tree
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At each node as we descend from the root
If the balance factor is nonzero
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If we descended in the direction of our
higher subtree
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If the next node down reports back that
the height of its subtree increased
This node has become unbalanced.
 We must do a rotation at this node.
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13
Rebalancing the Tree
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If we have to do a rotation
 If the next node down reports that it
descended in the same direction
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Do a single rotation.
Else
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Do a double rotation.
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Implementation
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Download:
http://www.cse.usf.edu/~turnerr/Data_Structures/Downloads/
2011_03_28_AVL_Tree/ File AVL_Tree_Demo.zip
Project contains:
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AVL_BST.h Our most recent BST template updated to
implement an AVL tree.
main.cpp Uses the AVL BST to replicate the states
example from last class.
Extract, build, and run
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Adding RI, PA, DE
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Adding GA, OH
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Adding GA, OH
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Adding MA
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Adding MA
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Adding MA
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Adding IL, MI, IN
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Adding IL, MI, IN
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Adding IL, MI, IN
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Adding NY
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Adding NY
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Adding NY
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Adding VT
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Adding VT
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Adding TX, WY
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Download

ppt