Trees (Ch. 9.2)
Longin Jan LateckiTemple University
based on slides by
Simon Langley and Shang-Hua Teng
Basic Data Structures - Trees
Informal: a tree is a structure that looks like a real tree (up-side-down)
Formal: a tree is a connected graph with no cycles.
Trees - Terminology
x
b e m
c d a
root
leaf
height=2
size=7
Every node must have its value(s)Non-leaf node has subtree(s)Non-root node has a single parent nodeA parent may have 1 or more children
value
subtree
nodes
Types of Tree
Binary Tree
m-ary Trees
Each node has at most 2 sub-trees
Each node has at most m sub-trees
Binary Search Trees
A binary search tree: … is a binary tree. if a node has value N, all values in its
left sub-tree are less than N, and all values in its right sub-tree are greater than N.
This is a binary search tree
This is NOT a binary search tree
5
4 7
3 2 8 9
Searching a binary search tree
search(t, s) {
If(s == label(t))
return t;
If(t is leaf) return null
If(s < label(t))
search(t’s left tree, s)
else
search(t’s right tree, s)}
h
Time per level
O(1)
O(1)
Total O(h)
Searching a binary search tree
search( t, s )
{ while(t != null)
{ if(s == label(t)) return t;
if(s < label(t)
t = leftSubTree(t);
else
t = rightSubTree(t);
}
return null;
h
Time per level
O(1)
O(1)
Total O(h)
Here’s another function that does the same (we search for label s):
TreeSearch(t, s)
while (t != NULL and s != label[t])
if (s < label[t])
t = left[t];
else
t = right[t];
return t;
Insertion in a binary search tree:we need to search before we insert
5
3 8
2 4 7 9
Time complexity ?
Insert 6 6
6
6
6
Insert 1111
11
11
O(height_of_tree)O(log n) if it is balanced n = size of the tree
always insert to a leaf
Insertion
insertInOrder(t, s)
{ if(t is an empty tree) // insert here
return a new tree node with value s
else if( s < label(t))
t.left = insertInOrder(t.left, s )
else
t.right = insertInOrder(t.right, s)
return t }
Try it!!
Build binary search trees for the following input sequences• 7, 4, 2, 6, 1, 3, 5, 7
• 7, 1, 2, 3, 4, 5, 6, 7
• 7, 4, 2, 1, 7, 3, 6, 5
• 1, 2, 3, 4, 5, 6, 7, 8
• 8, 7, 6, 5, 4, 3, 2, 1
Comparison –Insertion in an ordered list
Insert 6
Time complexity?
2 3 4 5 7 98
6 6 6 6 6
O(n) n = size of the list
insertInOrder(list, s) { loop1: search from beginning of list, look for an item >= s loop2: shift remaining list to its right, start from the end of list insert s}
6 7 8 9
Suppose we have 3GB character data file that we wish to include in an email.
Suppose file only contains 26 letters {a,…,z}. Suppose each letter in {a,…,z} occurs with frequency
f. Suppose we encode each letter by a binary code If we use a fixed length code, we need 5 bits for each
character The resulting message length is
Can we do better?
Data Compression
zba fff 5
Data Compression: A Smaller Example Suppose the file only has 6 letters {a,b,c,d,e,f}
with frequencies
Fixed length 3G=3000000000 bits Variable length
110011011111001010
101100011010001000
05.09.16.12.13.45.
fedcba
Fixed length
Variable length
G24.2405.409.316.312.313.145.
How to decode?
At first it is not obvious how decoding will happen, but this is possible if we use prefix codes
Prefix Codes No encoding of a
character can be the prefix of the longer encoding of another character:
We could not encode t as 01 and x as 01101 since 01 is a prefix of 01101
By using a binary tree representation we generate prefix codes with letters as leaves
e
a
t
n s
0 1
1
1
1
0
0
0
Prefix codes allow easy decoding
e
a
t
n s
0 1
1
1
1
0
0
0
Decode:
11111011100
s 1011100
sa 11100
san 0
sane
Prefix codes
A message can be decoded uniquely.
Following the tree until it reaches to a leaf, and then repeat!
Draw a few more trees and produce the codes!!!
Some Properties
Prefix codes allow easy decoding An optimal code must be a full binary tree (a
tree where every internal node has two children)
For C leaves there are C-1 internal nodes The number of bits to encode a file is
ccfT TCc
length )()B(
where f(c) is the freq of c, lengthT(c) is the tree depth of c, which corresponds to the code length of c
Optimal Prefix Coding Problem
Input: Given a set of n letters (c1,…, cn) with frequencies (f1,…, fn).
Construct a full binary tree T to define a prefix code that minimizes the average code length
iT
n
i i cfT length )Average(1
Greedy Algorithms
Many optimization problems can be solved using a greedy approach• The basic principle is that local optimal decisions may be used to
build an optimal solution
• But the greedy approach may not always lead to an optimal solution overall for all problems
• The key is knowing which problems will work with this approach and which will not
We study• The problem of generating Huffman codes
Greedy algorithms A greedy algorithm always makes the choice that looks
best at the moment• My everyday examples:
• Driving in Los Angeles, NY, or Boston for that matter
• Playing cards
• Invest on stocks
• Choose a university
• The hope: a locally optimal choice will lead to a globally optimal solution
• For some problems, it works
Greedy algorithms tend to be easier to code
David Huffman’s idea
A Term paper at MIT
Build the tree (code) bottom-up in a greedy fashion
Each tree has a weight in its root and symbols as its leaves.
We start with a forest of one vertex trees representing the input symbols.
We recursively merge two trees whose sum of weights is minimal until we have only one tree.
Building the Encoding Tree
Building the Encoding Tree
Building the Encoding TreeBuilding the Encoding Tree
Building the Encoding TreeBuilding the Encoding Tree
Building the Encoding TreeBuilding the Encoding Tree