CIS 068

Welcome to CIS 068 !Lesson 12:

Data Structures 3

Trees

CIS 068

Overview

• Binary Trees

• Complete Trees

• Heaps

• Binary Search Trees

• Balanced Trees

CIS 068

Definitions

Node 0

Node 1 Node 2 Node 3

Node 4 Node 5 Node 6

leaves

root

Node 1,2,3 are children of root

Node 4 and 5 are siblings

Node 1 is parent ofNodes 4,5

Node 0 is ancestorof all other nodes

Nodes 1-6 are descendants

of node 0

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Binary TreesDef. (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 TreesDef.:• 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

complete incomplete incomplete

depth 1depth 2depth 3

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Complete TreesProperties:• A complete tree with depth n has at most 2n+1 – 1 elements• A complete tree with depth n has at least 2n elements• The index of the left child of node k is 2k+1, the index of the right

child of node is 2k+2

01 2

3

7

4 5 6

8

depth 1depth 2depth 3

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Complete TreesStorage of complete trees in arrays:

01 2

3

7

4 5 6

8

0 1 2 3 4 5 6 7 8

…

2k+1, 2k+2k=3

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HeapDef.:• 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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HeapWhat for ?• Sorting (HEAPSORT):

– Sort elements into heap structure

• How to – Insert ?

– Delete ?

• Order of magnitude ?

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• 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 !

Heap

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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.

Binary Search Trees

76

39 80

32 47 79

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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)

Binary Search Trees

76

39

80

32 37

89

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• 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 !

Binary Search Trees

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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.

Balanced Trees

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Remark:

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

Balanced Trees

12

8 18

5 11 17

4

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How to create balanced trees ?

Rotation !

Binary Search Trees

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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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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

AVL Trees

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Example 1: single rotation

Is a single rotation always the solution ?

AVL Trees

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Example 2: single rotation

Example 2: the left-heavy tree got a right-heavy tree !

AVL Trees

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Example 3: rotation of subtrees

• Balance is achieved by rotating the subtrees in ‘a certain way’

AVL Trees

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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

AVL Trees

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…to be continued

Trees