Trees

1
Trees

• B.Ramamurthy
• Chapter 9
• CSE116A,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 an ADT can be classified as linear and
  non-linear.
• We will study non-linear ADTs tree and binary
  trees in this discussion.

3
Topics for discussion

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

4
Tree Formal Definition

• A tree T is a finite, non-empty set of nodes,
• T r U T1 U T2 U T3 U Tn
• with the following properties
• 1. A designated node of the set r, is called the
  root of the tree
• 2. The remaining nodes are partitioned into n gt
  0 subsets T1, T2, .. Tn each of which is a tree.

5
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.

6
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.

7
More Definitions

• The degree of a node is the number of subtrees
  associated with that node.
• A node of degree 0 is the leaf node.
• In a binary tree the degree of each node except
  the leaf nodes is 2.
• Level of a node of tree is measured from its
  root, whereas height of a node is measured from
  its root to the leaf. Height of a leaf node is 0.

8
Basic Tree Anatomy
Level 0
Level 1
Level 2
9
N-nary Tree

• An n-nary tree T is a finite set of nodes with
  the following properties
• 1. Either the set id empty or
• 2. The set consists of a root, R, and exactly N
  distinct N-ary trees. That is, the remaining
  nodes are partitioned into Ngt 0 sunbsets T0, T1,
  .. TN-1, each of which is an N-ary tree such that
  T R, T0, T1, .. TN-1.

10
Observation

• An important concept used in the above definition
  is recursion. A tree is defined in terms of
  smaller trees.
• This concept of defining a term by smaller of the
  kind is called recursive definition.
• We will discuss recursion next class.

11
External and Internal Nodes

• Theorem 9.1 (in your text) An N-ary tree with n
  gt 0 internal nodes contains (N-1)(n1) external
  nodes.
• Theorem 9.2 Consider an N-ary tree T of height
  h gt 0. The maximum number of internal nodes in T
  is given by

(h1)
- 1
N
N - 1
12
Leaf nodes

• Theorem 9.3 Consider an N-ary tree of height h
  gt 0. The maximum number of leaf nodes in T is
• N

h
13
Binary Tree

• A binary tree T is a finite set of nodes with the
  following properties
• 1. Either the set is empty, T O. or
• 2. The set consists of a root, r, and exactly two
  distinct binary trees T L and TR T r, TR, TL.
• TL is called the left subtree and TR is called
  the right subtree.

14
Binary tree example
A - B / C
How do you traverse this tree?
15
Recursive Solutions

• Recursion is an important problem solving
  approach that is an alternative to iteration.
• These are questions to answer when using
  recursive solution
• 1. How can you define the problem in terms of a
  smaller problem of the same type?
• 2. How does each recursive call diminish the size
  of the problem?
• 3. What instance of the problem can serve as the
  base case?
• 4. As the problem size diminishes, will you reach
  this base case?

16
Example 1 Factorial of N

• Iterative definition
• Factorial(N) N (N-1) (N-2) 1 for any N gt
  0.
• Factorial(0) 1
• Recursive solution
• 1. Factorial(N) N Factorial(N-1)
• 2. Problem diminishing? yes.
• 3. Base case Factorial(0) 1 base case does
  not have a recursive call.
• 4. Can reach base case as problem diminishes? yes

17
Writing String Backwards

• 1. Write a string of length N backwards in terms
  of writing a string of length (N-1) backwards.
• 2. Base case
• Choice 1 Writing empty string.
• Choice 2 Writing a string of length 1.

18
WriteBackward(S)

• if string is empty
• do nothing // this is the base case
• else
• write the last character of S
• WriteBackward(S - minus its last character

19
Defining Languages

• A grammar states the rules of a language.
• A recursive algorithm can be written based on
  this grammar that determines whether a given
  string is a member of the language gt
  recognition algorithm.
• For expressing a grammar we use special symbols
• X Y means X or Y
• X.Y or simply X Y means X followed by Y
• ltxyzgt an instance of entity xyz

20
A grammar for Java identifiers

• ltidentifiergt ltlettergt ltidentifiergt ltlettergt
  ltidentifiergtltdigitgt
• ltlettergt a b z A B CZ_
• ltdigitgt 0 1 9

21
Recognition algorithm for identifiers

• boolean IsId (w)
• // Returns TRUE if w is a legal Java
  identifier,
• // Otherwise returns FALSE
• if (w is of length 1)
• if (w is a letter)
• return true
• else
• return false
• else if (last char of w is a letter or a digit)
• return IsId(w minus its last char)
• else return false

22
Tree Traversals

• Preorder, inorder and post order.
• Pre, In and Post refer to the order in which root
  is traversed.
• PreOder traversal Visit root, traverse TL,
  Traverse TR.
• InOrder traversal Traverse TL, visit root,
  traverse TR.
• PostOrder traversal Traverse TL, Traverse TR,
  and visit root.

23
Expression Trees

/

A
B
--
E
D
C
24
Implementing a Binary Tree

• Examine Java API
• Use the existing classes.
• Extend existing classes.
• Implement necessary interfaces.

25
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.

26
Example Tree of names
Jane
Bob
Tom
Ellen
Alan
Wendy
Nancy
Where will you insert the name Randy? How about
Ian?