Trees

1
Trees

• B.Ramamurthy
• Chapter 10
• CS114A,B

2
Introduction

• Data organizations we studied are linear in that
  items are one after another.
• Data organization can also be hierarchical
  whereby an item can have more than one immediate
  successor.
• Thus ADT can be classified as linear and
  non-linear.
• We will study non-linear ADTs tree, binary tree
  and binary search tree and graph in this
  discussion.

3
Topics for discussion

• Terminology
• Binary Tree -- Formal definition
• Binary Search tree
• Tree Traversal
• Full binary tree
• Complete binary tree
• Balanced Tree
• Summary

4
Terminology

• Trees are used to represent relationships items
  in a tree are referred to as nodes and the lines
  connecting the nodes that express the
  hierarchical relationship are referred to as
  edges.
• The edges in a tree are directed.
• Trees are hierarchical which means that a
  parent-child relationship exist between the nodes
  in the tree.
• Each node has at most one parent. The node with
  no parents is the root node.
• The nodes that have no successors (no children
  nodes) are known as leaf nodes.

5
Basic Tree Anatomy
Level 0
Level 1
Level 2
6
Binary Tree formal definition

• A binary tree is a set T of nodes such that
  either
• T is empty, or
• T is parameterized into three disjoint sets
• A single node r, the root
• Two possibly empty sets that are binary trees
  called left and right subtrees of r.
• Example Binary trees to represent algebraic
  expressions.

7
Tree Types

• Binary tree
• Full binary tree
• Complete binary tree
• Binary Search Tree

8
Binary tree example
A - B / C
A
\
C
B
How do you traverse this tree?
9
Binary Search Tree

• A binary search tree is a binary tree that is in
  a sense sorted according to the values in its
  nodes.
• For each node n, a binary search tree satisfies
  the following three properties
• 1. ns value is greater than all values in its
  left subtree TL.
• 2. ns value is less than all its values in its
  right subtree TR.
• 3. Both TL and TR are binary search tree.

10
Example Tree of names
Jane
Bob
Tom
Ellen
Alan
Wendy
Nancy
Where will you insert the name Randy? How about
Ian?
11
Hwk3 Q1
12
Hwk3 Q2
Due Date 4/9/99
13
Types of Trees
trees
dynamic
static
game trees
search trees
priority queues and heaps
graphs
Huffman coding tree
14
Complete Binary Tree

• A complete binary tree is a binary tree
• with leaves on either a single level or on two
  adjacent levels
• such that the leaves on the bottommost level are
  placed as far left as possible
• such that nodes appear in every possible position
  in each level above the bottom level.

15
Example
Complete Binary Tree
Incomplete Binary Tree
16
Example (contd.)
Complete Binary Tree
17
A Contiguous Representation for a complete binary
tree
1
Lets number the nodes starting from the
root, level by level, left to right
2
3
5
4
7
6
8
9
11
10
12
18
Finding nodes in Contiguous Representation
Let Aj be a node and n be the number of
nodes. Left child of Aj is A2j if 2j lt
n Right child of Aj is A2j1 id 2j1 lt
n Parent of Aj is Aj/2 if j gt 1 Root is
A1 If 2j gt n the Aj is a leaf. Lets look
at an example (from last years final exam)
19
Exercise
Given a complete binary tree with 2670 nodes and
with contiguous representation A1 to
A2670 what is the maximum level l of the
tree? How many nodes are in the level n where n lt
l ? Give an expression in terms of n. Derive the
expression for number of internal nodes of the
tree. How many leaf nodes are there in this tree?
Where is the parent of node A261
located? Where are the children of node at A261
located?
20
Exercise (contd.)
21
Heap A Formal Definition

• Heap is a loosely ordered complete binary tree.
• A heap is a complete binary tree with values
  stored in its nodes such that no child has a
  value greater than the value of its parent.
• A heap is a complete binary tree
• 1. That is empty or
• 2a. Whose root contains a search key greater than
  or equal to both its children node.
• 2b. Whose left subtree and right subtree are
  heaps.

22
Types of heaps

• Heaps can be max heaps or min heaps. Above
  definition was for a max heap.
• In a max-heap the root is higher than or equal to
  the values in its left and right child.
• In a min-heap the root is smaller than or equal
  to the values in its left and right child.

23
ADT Heap

• createHeap ( )
• destroyHeap ( )
• empty ( )
• heapInsert (NewItem)
• heapDelete ( RootNode)

24
Example
Consider data set
25
Example
1
26
Delete Root Item

• In a max heap the root item is the largest and is
  the chosen one for deletion.
• 1. After deletion of root, two disjoint heaps
  result.
• 2. Place last node as a root, to form a
  semi-heap.
• 3. Use trickle-down to form a heap.
• Growth rate function 3log N 1 O(log N)
• Consider a heap example discussed above.
• Delete root item.

27
Insert An Item

• 1. Insert a node as a last node.
• 2. Trickle up (repeat for various levels) to form
  a heap.
• Consider inserting 15 into the heap of the
  delete example.
• Insert is also a O(log N) operation.

28
Insert Node Example
15
5
9
3
6
2
29
Priority Queue

• Priority Queue implemented as a heap. Priority
  queue inserts are done according to some criteria
  known as priority
• Insert location are according to priority and
  delete are to the head of queue.
• PQueue constructor
• PQueueInsert
• PQueueDelete
• Data
• Heap PQ

30
Heap Sort

• Heapsort
• 1. Represent elements as a max-heap.
• 2. Delete item at root, as it is the largest.
• 3. Heap, repeat step 2, until one item is left.
• O(Nlog N) in the worst case!