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Trees
Lecture 54
Section 11.5
Fri, Apr 28, 2006
Trees
A graph is circuit-free (or acyclic) if it has
no nontrivial circuits.
 A tree is a connected circuit-free graph.

Examples of Trees
A directory structure on a hard drive is a
tree structure.
 Binary search tree.
 Expression tree.
 Huffman tree for data compression.
 Parse tree in a compiler.

Directory Trees
My Documents
My Music
My Pictures
My Homework
Math 262
Coms 262
Binary Search Trees
50
30
10
75
40
85
Expression Trees
*

+
10
15
25
Huffman Trees
Consider the message
ATTACK AT DAWN
 Count the frequencies of the letters.

A–4
T–3
C–1
D–1

K–1
N–1
W–1

Huffman Trees
12
7
4
A
5
3
T
3
2
2
1
W
1
1
C
D
1
K
1
N
Huffman Trees
0
0
A
1
1
0
0
T
0
C
1
D
1
1
0
1
W
K
N
Huffman Trees

Encoding scheme:
A = 00
 T = 01
 C = 1000
 D = 1001
 K = 110
 N = 111
 W = 101

Huffman Trees

Encoding of the message ATTACK AT
DAWN:
0001010010001100001100100101111.
 A total of 31 bits.
 ASCII would require 96 bits (12 chars).
 Compression ratio = 31/96 < 1/3.

Parse Trees
Consider the program statement
if (x > 0) y = 1
 The statement includes the following
“terminals.”

Keyword if
 Parentheses ( and )
 Relative operators > and =
 Variables x and y
 Numbers 0 and 1

Parse Trees

Parsing the statement uses the following
“grammar” rules.
stmt if ( rel-expr ) stmt
 stmt  expr ;
 expr  var = expr

expr  num
 rel-expr  expr rel-op expr
 rel-op  >

Parse Trees
stmt
if
(
rel-expr
expr
rel-op
expr
var
>
num
var
0
y
x
stmt
)
expr
;
=
expr
num
1
Tree Characteristics
Theorem: A tree must have at least one
vertex of degree 1.
 Proof:

Suppose not.
 Then each edge has degree at least 2.
 Using the idea in the proof for Euler circuits,
if we pursue a walk, we will hit a dead-end
only at a previously visited vertex.

Proof
This contradicts the property that trees are
circuit free.
 Therefore, there must exist a vertex of
degree 1.

Tree Characteristics
Theorem: If a tree has n vertices, then it
has exactly n – 1 edges.
 Proof by induction.
 Base case: n = 1.


The tree has 1 vertex and 0 edges, so the
statement is true when n = 1.
Proof

Inductive case:
Suppose the statement is true when n = k
for some integer k  1.
 Let T be a tree with k + 1 vertices.
 Let v be a vertex of T of degree 1.
 Create a new graph T by removing v and
its one incident edge.
 Then T is a tree with k vertices.

Proof
By the induction hypothesis, T has k – 1
edges.
 Therefore, T has k edges.


Therefore, the statement is true for all n 
1.
Tree Characteristics

Theorem: Let G be a graph with n vertices.
If G has any two of the following three
properties, then it has all three properties
and G is a tree.
G is circuit free.
 G is connected.
 G has n – 1 edges.

Rooted Binary Trees
A binary tree is rooted if it has one node
designated the root.
 The other nodes are arranged in levels,
depending on their distance from the root.

Level 0 – The root.
 Level 1 – Nodes adjacent to the root.
 Level n – Nodes adjacent to level n – 1
nodes (but not the level n – 2 nodes).

Counting Rooted Binary
Trees
How many rooted binary trees are there
with n vertices?
 The tree must have 1 root vertex.
 That leaves n – 1 vertices for the left and
right subtrees.

Counting Rooted Binary
Trees

They can be divvied up in n ways:
0 on left, n – 1 on right.
 1 on left, n – 2 on right.
 2 on left. n – 3 on right.
:
 n – 1 on left, 0 on right.

Counting Rooted Binary
Trees

In each case, the number of such binary
trees is the number of left subtrees of size
k times the number of right subtrees of size
n – k – 1.
Counting Rooted Binary
Trees
Let Cn be the number of binary trees of size
n.
 Then

C0  1 (by definition )
n 1
Cn   Ck Cn  k 1 , for all n  1.
k 0
Counting Rooted Binary
Trees
The first five terms are 1, 2, 5, 14, 41.
 The 6th term is
141 + 214 + 55 + 142 + 411 = 163.
 Is there a non-recursive formula?

The Catalan Numbers
These numbers turn out to be the Catalan
numbers.
 The non-recursive formula is

 2n 
 
n

Cn 
n 1
Restricted Walks

How many walks are there from A to B, by
going only east and north?
B
N
A
Counting Unrooted Trees
How many non-isomorphic unrooted trees
are there with n vertices?
 n = 5:
