Binary Trees

1
Binary Trees

• Computer Science and Engineering

2
Introduction

• We studied linked list is a dynamic linear data
  structure.
• It is dynamic since it supports efficient
  addition and deletion of items.
• It is linear since it is sequential and each
  element in it has exactly one successor.
• A tree is nonlinear data structure.
• Each element may have more than one successor.
• Can be static or dynamic.

3
Topics for Discussion

• Elements of a tree
• Examples of trees
• Binary Tree Definition
• Types of binary trees
• Contiguous (static) representation
• Dynamic representation

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.
• Lets look at some examples and identify the
  various elements.

5
Examples

• Family ancestor tree
• Directory of files organization in your computer
  system
• Parse tree
• Languages are defined using grammar
• Grammars are specified using rules or syntax
• Syntax is expressed using a notation called
  Backaus-Naur Form (BNF) (John Backus and Peter
  Naur)
• Expression trees
• Game trees

6
An Ancester Tree
(From Greek mythology)

Gaea
Cronus
Phoebe
Ocean
Zeus
Poseidon
Demeter
Pluto
Leto

Apollo
7
BNF for a Language

• BNF notation includes nonterminals and terminals.
  
• Terminals are literals or particular symbols.
• Nonterminals are general expressions that can be
  substituted with terminals and nonterminals.
  Grammar rules specify the definition of a
  nonterminal. Nonterminals are enclosed with angle
  brackets ltnonterminalgt
• Symbols used in construction include
  (defines), (or) and other common operators.

8
BNF for a Java Statement
ltstatementgt ltselection-stmtgt
ltother-stmtgt ltselection-stmtgt if (ltexprgt)
ltstatementgt else
ltstatementgt ltexprgt ltrelational-exprgtltassign-ex
prgtltidentifiergt ltrelational-exprgt ltexprgt
ltrel-opgt ltexprgt ltassign-exprgt ltexprgt
ltexprgt

9
Parse tree
ltstatementgt
ltselection-stmtgt
if
(
ltexprgt
)
ltstatementgt
else
ltstatementgt

ltexprgt
ltrelational-exprgt
ltexprgt
ltexprgt
ltrel-opgt
.
A major task of the compiler is to construct a
parse tree from the input program and verify it
is correct.
ltidentifiergt
ltidentifiergt
lt
b
a
10
Expression tree

A B C D

ltleftgtltrootgtltrightgt (in-order expression) ltrootgtlt
leftgtltrightgt (pre-order expressiongt ltleftgtltrightgt
ltrootgt (post-order expression)

A
B
C
D
Single representation Multiple views
11
Game Tree

X
X
X
.

X
X
X
X
X
X
12
Binary Tree

• A binary tree can be defined recursively as
  follows. It is either
• empty, or
• consists of a root node together with left and
  right trees, both of which are binary trees.

13
Binary Tree
NonEmpty
Empty
NullObject (pattern) Singleton (pattern)
14
Binary Tree (contd.)
15
Binary Tree (contd.)
16
Characteristics of trees

• A path is a sequence of nodes n1, n2, ..., nk
  such that node ni is the parent of node ni1 for
  all 1 lt i lt k.
• The length of a path is the number of edges on
  the path.
• The height of a node is the length of the longest
  path from the node to a leaf.
• The height of tree is the height of its root.
• The level of a node is the length of the path
  from the root to the node.

17
Full Binary Tree

• A full binary tree is a tree in which each node
  has exactly zero or two non-empty children. All
  leaves are at the same level.
• A complete binary tree in which all the leaf
  nodes are in level n or n-1 and all leaves on the
  level n are filled from left to right.
• There are some interesting properties that arise
  out of this definition.
• Lets look at some examples to illustrate the
  various definitions.

18
Example
root
Level 0
Level 1
internal node
Height of the tree3
leaf
19
Contiguous Representation for complete binary tree
1
2
3
5
4
6
7
8
20
Complete binary tree (contd.)

• Number the N nodes sequentially, from 1.. N,
  starting with the root , level by level from left
  to right.
• Parent(n) floor(n/2) for ngt1
• Left child(n) 2n (if 2n lt N, else no left
  child)
• Right child(n) 2n1 (if 2n1 lt N, else no
  right child)
• The node number of children and parent can be
  calculated for a given node n.

21
Contiguous representation

• By placing the N nodes in a contiguous sequence
  we can use simple arithmetic relationships to
  process the nodes. This will eliminate storage
  for the pointers to sub trees.

Root object
1
2
3
4
5
6
8
0
7
22
Array Representation

• Refer to the array in slide 21 which represents
  the complete binary tree in slide 19
• Array index 1 has the root object, 2 and 3 the
  left and right sub tree of root object
  respectively and so on.
• The storage needed for the pointers to the left
  and right sub tree for each node is eliminated
• But the location need to be calculated every time
  a node is accessed. Trade off is between the
  storage need for the pointers and extra execution
  time incurred for computing the location.

23
Linked Representation

• In its simplest form
• class BTree
• Object obj
• BTree left
• BTree right
• //constructor get, set methods
• //public Object visitor() //for all other
  //operations

24
Linked Representation

• Simple interface/implementation
• /application
• Addition of a visitor (Visitor Pattern)
• State-based implementation