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General Info. Check out gilgarn.org. QUIZ. DO YOU: Love CSC 171 ? Want a job? Like to exert power over others? Want to improve CSC education @ UR? DID YOU: Do well (A or B) in 171? Think Workshops were helpful? CAN YOU: Stand looking at me for another semester?. IF YOU:.

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general info
General Info
  • Check out gilgarn.org
slide2
QUIZ
  • DO YOU:
    • Love CSC 171 ?
    • Want a job?
    • Like to exert power over others?
    • Want to improve CSC education @ UR?
  • DID YOU:
    • Do well (A or B) in 171?
    • Think Workshops were helpful?
  • CAN YOU:
    • Stand looking at me for another semester?
if you
IF YOU:
  • Answered “YES” to most of the above questions
  • THEN:
    • Consider serving as a WORKSHOP LEADER
  • BENEFITS:
    • FAME, MONEY, POWER
    • LOOKS GOOD ON YOUR RESUME
    • DO SOMETHING GOOD FOR HUMANITY
  • INTEREST MEETING:
    • THURS (3/25) 5:15PM
    • LAS, basement of Lattimore
trees

Trees

CSC 172

SPRING 2004

LECTURE 14

lists
Lists

We have seen lists:

public class Node {

Object data;

Node next;

}

trees6

Trees

Now, look at trees:

public class Node {

Object data;

Node left;

Node right;

}

rooted trees
Rooted Trees

Collection of nodes, one of which is the root

Nodes != root have a unique parent node

Each non-root can reach the root by following parent links one or more times

definitions
Definitions

If node p is the parent of node c

then c is a child of p

Leaf : no children

Interior node : has children

Path : list of nodes (m1,m2,…,mk) such that each is the parent of the following

path “from m1 to mk”

Path length = k-1, number of links, not nodes

slide9
If there is a path from m to n, then m is an ancestor of n and n is a descendant of m

Note m == n is possible

Proper ancestors, descendants : m != n

Height of a node n is the length of the longest path from n to a leaf

Height of a tree is the height of its root

Depth of a node is the length of the path from the root to n

Subtree rooted at n is all the descendants of n

slide10
The children of any given note are often ordered “from the left”

Child c1 is to the left of c2 then all the nodes in the subtree rooted at c1 are “to the left” of those in the subtree rooted at c2

Nodes may have labels, which are values associated with the nodes

example unix file systems

/bin

/dev

/usr

.

/dev/term

/dev/sound

/dev/tty01

/usr/anna

/usr/jon

/usr/ted

Example: UNIX File Systems

/

example expression trees
Example: Expression Trees

Labels are operands or operators

Leaves : operands

Interior nodes : operators

Children are roots of sub-expressions to which the operator is applied

recursion on trees
Recursion on Trees

Many algorithms to process trees are designed with a basis (leaves) and induction (interior nodes)

Example: If we have an expression tree we can get

infix

(operator between operands - common)

prefix

(operator before operands – like function calls)

postfix

(operator after operands – good for compilers)

expression tree to postfix
Expression Tree to Postfix

Basis

For a leaf, just print the operand

Induction:

For an interior node

apply algorithm to each child from left

print the operator

x 1 x y 416
*

+

-

4

x

1

-

x

y

(x+1)*(x-y+4)

x 1 + x y – 4 - *

general trees

General Trees

Some trees are binary:

public class Node {

Object data;

Node left;

Node right;

}

some trees are not binary

/bin

/dev

/usr

.

/dev/term

/dev/sound

/dev/tty01

/usr/anna

/usr/jon

/usr/ted

Some trees are not binary

/

How do we implement

such trees?

lmc rs
LMC-RS

Leftmost-Child, Right-Sibling Tree Representation

Each node has a reference to

  • It’s leftmost child
  • It’s right sibling – the node immediately to the right having the same parent

Advantage: represents trees without limits or pre-specified number of children

Disadvantage: to find the ith child of node n, you must traverse a list n long

lmc rs20

LMC-RS

public class Node {

Object data;

Node l_child;

Node r_sibling;

}

structural induction
Structural Induction

Basis = leaves (one-node trees)

Induction = interior nodes (trees with => 2 nodes)

Assume the statement holds for the subtrees at the children of the root and prove the statement for the whole tree

tree proof example

Tree Proof Example

Consider a LMC-RS tree

S(T): T has one more  reference than it has nodes

Basis: T is a single node – 2  references

induction
Induction

T has a root r and one or more sub trees T1, T2,…,Tk

BTIH: each of these trees, by itself has one more  than nodes

How many nodes all together?

How many  references?

How many nodes do I add to make one tree?

How many  references do we reduce to make one tree?

slide24
T1

n1 nodes

n1+1 

T2

n2 nodes

n2+1 

Tk

nk nodes

nk+1 

?

?

?

One more node

One more 

Still “k” extra

slide25
T1

n1 nodes

n1+1 

T2

n2 nodes

n2+1 

Tk

nk nodes

nk+1 

?

?

?

One more node

One more 

Still “k” extra

How many less?

example pair quiz
Example: Pair Quiz

S(T): A full binary tree of height h

has (2h+1 – 1) nodes

Basis?

Induction?

example
Example:

S(T): A full binary tree of height h has 2h+1 – 1 nodes

Basis?

Nodes = 1, height == 0, 20+1-1 = 1

Induction?

slide28
T1

h height

2 h+1-1 nodes

T2

h height

2 h+1-1 nodes

Height = h+1

why structural induction
Why Structural Induction?
  • Q: Why do I want you to draw the pictures?
  • A: We know you can do the algebra. As you move on, the difficult part of the proofs is “setting up” the proof, from some situation. Sure, if someone sets it up for you, it’s just an “exercise” to solve. The real value is in being able to reason about a situation so as to set up a proof.
process
Process

Real world

phenomena

or artifact

Formal

Representation

“set up” of proof

Rigorous

Knowledge

Formal

techniques:

induction,

algebra

Informal

reasoning

abstraction

analysis

kraft s inequality
Kraft’s Inequality
  • Suppose that a binary tree has leaves {l1,l2,..lM} at depths {d1,d2,...,dM}, respectively.
  • Prove:
how do you set this up33
How do you set this up?
  • Basis:
  • In a tree with zero nodes, the sum is zero
  • In a tree with one node
    • One leaf, depth 0
slide34
T1

∑1<=1

T2

∑2<=1

Same number of total leaves, all depths increase by 1

So, in new Tree ∑1<=1/2 && ∑2<=1/2

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