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Trees
The definitions for this presentation are from from:
Corman , et. al., Introduction to Algorithms (MIT Press),
Chapter 5.
Some material on binomial trees is from Hull.
Data Structures and Algorithms
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A Few Applications
Arithmetic Expressions
+
b+a*b
b
*
a
Data Structures and Algorithms
b
2
A Few Applications
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Data Structures and Algorithms
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Object Model
3
A Few Applications
Binomial “Trees” Movements
Binomial trees are
in Time dt
frequently used to
Su
S
Sd
From Hull
approximate the
movements in the
price of a stock or
other asset
In each small interval
of time the stock price
is assumed to move up
by a proportional amount
u or to move down by a
proportional amount d.
4
Definitions
A Free tree is a connected, acyclic undirected graph.
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If an undirected graph is acyclic but possibly disconnected,
it is a forest.
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This is a graph that contains a cycle and is therefore neither a
tree nor a forest.
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Theorem (Properties of Free Trees)
Let G = (V, E) be an undirected graph.
The following statements are equivalent:
1. G is a free tree.
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2. Any two vertices in G are connected by a unique simple
path.
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3. G is connected, but if any edge is removed from E, the
resulting graph is disconnected.
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4. G is connected, and |E| = |V| - 1.
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5. G is acyclic, and |E| = |V| - 1.
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6. G is acyclic, but if any edge is added to E, the resulting
graph contains a cycle.
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A rooted tree is a free tree in which one of the vertices is
distinguished from the others. The distinguished vertex is
called the root of the tree. We often refer to a vertex of a
rooted tree as a node (we may also call this a vertex) of the
tree. The following figure shows a rooted tree on a set of 12
nodes with root 7.
7
3
8
6
10
12
5
4
11
2
1
9
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Consider a node x in a rooted tree T with root r. Any
node y on the unique path from r to x is called an ancestor
of x. If y is an ancestor of x, then x is a descendant of y.
(Every node is both an ancestor and a descendant of itself.)
If y is an ancestor of x and x  y, then y is a proper ancestor
of x and x is a proper descendant of y. The subtree rooted
at x is the tree induced by descendants of x, rooted at x.
r
8
y
5
6
9
x
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If the last edge on the path from the root r of a tree T to a
node x is (y, x), then y is the parent of x, and x is a child of
y. The root is the only node in T with no parent. If two
nodes have the same parent, they are siblings. A node with
no children is an external node or leaf. A nonleaf node is
an internal node.
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The number of children of a node x in a rooted tree T is
called the degree of x. The length of the path from the
root r to a node x is the depth of x in T. The largest depth
of any node in T is the height of T.
7
3
depth 0
10
depth 1
4
height = 4
8
6
12
5
11
1
2
depth 2
depth 3
depth 4
9
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An ordered tree is a rooted tree in which the children of
each node are ordered. That is, if a node has k children,
then there is a first child, a second child, …, and a kth
child. The two trees shown below are different when
considered to be ordered trees, but the same when
considered to be just rooted trees.
7
7
3
8
6
11
12
5
4
10
1
3
12
2
1
9
10
8
4
11
6
2
5
9
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A binary tree T is a structure defined on a finite set of
nodes that either
• contains no nodes, or
• is comprised of three disjoint sets of nodes:
a root node, a binary tree called its left subtree
and a binary tree called its right subtree.
3
2
7
4
1
5
6
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Full binary tree: each node is either a leaf or has degree
exactly 2.
In a positional tree, the children of a node are labeled
with distinct positive integers. The ith child of a node is
absent if no child is labeled with integer i. A k-ary tree
is a positional tree in which for every node, all children
with labels greater than k are missing. Thus, a binary
tree is a k-ary tree with k = 2.
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A complete k-ary tree is a k-ary tree in which all leaves
have the same depth and all internal nodes have degree
k.
depth 0
depth 1
depth 2
height = 3
depth 3
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How many leaves L does a complete binary tree of height h
have?
d=0
d= 1
d=2
The number of leaves at depth d = 2d
If the height of the tree is h it has 2h leaves.
L = 2h.
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What is the height h of a complete binary tree with L leaves?
leaves = 1
height = 0
leaves = 2
height = 1
leaves = 4
height = 2
leaves = L
height = Log2L
Since L = 2h
Log2L = Log22h
h = Log2L
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The number of internal nodes of a complete binary
tree of height h is ?
1 + 2 + 22 + . . . + 2 h-1 =  2i
Internal nodes = 0
height = 0
Internal nodes = 1
height = 1
Internal nodes = 1 + 2
height = 2
Internal nodes = 1 + 2 + 4
height = 3
Geometric series
= 2h - 1
2-1
Thus, a complete binary tree of height = h has 2h-1 internal
nodes.
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The number of nodes n of a complete binary
tree of height h is ?
nodes = 1
height = 0
nodes = 3
height = 1
nodes = 7
height = 2
nodes = 2h+1- 1
height = h
Since L = 2h
and since the number of internal nodes = 2h-1 the
total number of nodes n = 2h+ 2h-1 = 2(2h) – 1 = 2h+1- 1.
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If the number of nodes is n then what is the height?
nodes = 1
height = 0
nodes = 3
height = 1
nodes = 7
height = 2
nodes = n
height = Log2(n+1) - 1
Since n = 2h+1-1
n + 1 = 2h+1
Log2(n+1) = Log2 2h+1
Log2(n+1) = h+1
h = Log2(n+1) - 1
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Catalan Numbers
1  2n 
 
n 1 n 
=
1
(2n)!
(n+1) n! (2n-n)!
1,1,2,5,14,...
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The number of distinct binary
trees with n nodes
N=0
N=1
N=2
1  2n 
 
n 1 n 
N=3
...
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Class for Binary Nodes
public class BTNode
{
private Object data;
private BTNode left;
private BTNode right;
...
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public BTNode(Object obj,
BTNode l,
BTNode r)
{
data = obj; left = l; right= r;
}
public boolean isLeaf()
{
return (left == null) &&
(right == null);
}
...
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Copying Trees
public static BTNode
treeCopy(BTNode t)
{
if (t == null) return null;
else
{
BTNode leftCopy =
treeCopy(t.left);
BTNode rightCopy =
treeCopy(t.right);
return new BTNode(t.data,
leftCopy,
rightCopy);
} Data Structures and Algorithms
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Tree Traversals
•Preorder
•Inorder
•Postorder
•Levelorder
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public void preOrderPrint(){
System.out.println(data);
if (left != null)
left.preOrderPrint();
if (right != null)
right.preOrderPrint();
}
a b c d e f g
a
e
b
c
d
f
g
Root, Left, Right
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public void inOrderPrint(){
if (left != null) {
left.inOrderPrint()
System.out.println(data);
if (right != null)
right.inOrderPrint()
}
c b d a f e g
a
b
Left, Root, Right
c
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d f
g
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public void postOrder(){
if (left != null
left.postOrder()
if (right != null)
right.postOrder()
System.out.println(data);
}
a
b
e
c d b f g e a
c
d
f
g
Left, right, root
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levelorder (T) {
Q = makeEmptyQueue()
enqueue (T,Q)
until isempty (Q) {
p = dequeue(Q)
visit (p)
for each child  of P, in order, do
enqueue (, Q)
a
}
}
b
a b e c d f g
Data Structures and Algorithms
c
d
e
f
g
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An Array Representation
Suppose we are lucky enough to be working with complete
binary trees.
We can store the tree in an array.
Let the root be at index 0 and let the left and right children
of node i be at indexes 2i+1 and 2i+2 respectively.
Lewis and Denemberg, Page 111
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An Array Representation
isLeaf(i) : 2i + 1 >= n
leftChild(i) : 2i + 1 (none if 2i+ 1 >= n)
rightChild(i): 2i + 2 (none if 2i + 2 >= n)
leftSibling(i): i - 1 (none if i == 0 or i is odd)
rightSibling(i) : i + 1 (none if i = n-1 or i is even)
parent(i) = int((i-1)/2) (none if i == 0)
Works if the tree is “almost complete”, growing top to bottom
and left to right.
Lewis and Denemberg, Page 111
Data Structures and Algorithms
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Binomial “Trees” Using an Array
Representation
S0u 3
S0u 4
S0u
S0u 2
S0u 2
S0u
S0
S0d
S0
S0d
S0d 2
The array element bt[0] will be S0.
In general, given a node with index i at depth d,
its left child is located at bt[i + d+1] and
its right child is located at bt[i + d+2]
Data Structures and Algorithms
S0d 3
S0
S 0d 2
S 0d 4
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