Welcome to CIS 068

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Welcome to CIS 068 !
Lesson 12 Data Structures 3 Trees
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Overview

• Binary Trees
• Complete Trees
• Heaps
• Binary Search Trees
• Balanced Trees

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Definitions
root
Node 0 is ancestor of all other nodes Nodes 1-6
are descendants of node 0
Node 0
Node 1,2,3 are children of root
Node 1
Node 2
Node 3
Node 1 is parent of Nodes 4,5
Node 4
Node 5
Node 6
Node 4 and 5 are siblings
leaves
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Binary Trees

• Def. (recursively defined data structure)
• A binary tree is either
• empty (no nodes), or
• has a root node, a left binary tree, and a right
  binary tree as children
• hence it is a tree with at most two children
  for each node

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Complete Trees

• Def.
• A tree in which all leaf nodes are at some depth
  n or n-1, and all leaves at depth n are toward
  the left

depth 1 depth 2 depth 3
complete
incomplete
incomplete
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Complete Trees

• Properties
• A complete tree with depth n has at most 2n1 1
  elements
• A complete tree with depth n has at least 2n
  elements
• The index of the left child of node k is 2k1,
  the index of the right child of node is 2k2

0
depth 1 depth 2 depth 3
1
2
3
4
5
6
7
8
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Complete Trees

• Storage of complete trees in arrays

0
1
2
3
4
5
6
7
8

0
1
2
3
4
5
6
7
8
2k1, 2k2
k3
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Heap

• Def.
• A complete binary tree where every node has a
  value greater than or equal to the key of its
  parent

89
76
80
37
32
39
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Heap

• What for ?
• Sorting (HEAPSORT)
• Sort elements into heap structure
• How to
• Insert ?
• Delete ?
• Order of magnitude ?

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Heap

• Example of Heapsort HEAPSORT-APPLET
• Insert / Delete (board)
• Heaps provide a structure for efficient retrieval
  of maximum values !
• How to look for arbitrary values ? Binary Search
  Trees !

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Binary Search Trees

• Def.
• A binary tree where every node's left subtree has
  values less than the node's value, and every
  right subtree has values greater.

76
39
80
32
47
79
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Binary Search Trees

• Remarks
• A heap is NOT a binary search tree !
• A binary search tree is not necessarily complete
  (see example)!
• (Worst case create BST of sorted list)

89
80
76
39
32
37
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Binary Search Trees

• How to search in binary search tree ?
• (Answer is straightforward)
• Applet animated BST
• Order of magnitude ?
• How to achieve O(log n) ?
• balanced binary search trees !

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Balanced Trees

• Def.
• A tree whose subtrees differ in height by no more
  than one and the subtrees are height-balanced,
  too. An empty tree is height-balanced.

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Balanced Trees

• Remark
• Every complete tree is balanced, but not vice
  versa !

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8
5
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4
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Binary Search Trees

• How to create balanced trees ?
• Rotation !

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Rotation
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Rotation
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Rotation
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AVL Trees

• How to use Rotation to create balanced trees
• AVL Trees
• (Adelson-Velskii Landis, 1962)

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AVL Trees

• Idea
• Keep track of the difference in the depth of each
  subtree as items are inserted or removed
• Use rotation to bring tree into balance if
  difference is not in range of -1

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AVL Trees

• Example 1 single rotation
• Is a single rotation always the solution ?

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AVL Trees

• Example 2 single rotation
• Example 2 the left-heavy tree got a right-heavy
  tree !

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AVL Trees

• Example 3 rotation of subtrees
• Balance is achieved by rotating the subtrees in
  a certain way

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AVL Trees

• Resume
• Special binary trees provide an efficient
  structure for sorting and searching
• Complete binary trees can be stored in an array
  without link-structure overhead
• Heaps are used to sort arrays for retrieval of
  maximum values (typical application
  shape-abstraction- assignment !)
• Binary search trees are used for access to
  arbitrary objects in O(log n), achieved by
  balancing trees
• AVL trees are one example for balanced trees,
  using rotation to keep the balance

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Trees

• to be continued