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

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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.

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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.

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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.

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Example
root
Level 0
Level 1
internal node
Height of the tree3
leaf