CSI2110

Data Structures and Algorithms

Prof. WonSook Lee

1
 Trees

• Trees
• Binary Trees
• Properties of Binary Trees
• Traversals of Trees
• Data Structures for Trees

2
a Tree

3
 Trees
A graph G = (V,E) consists of an set V of VERTICES
and a set E of edges, with E = {(u,v): u,v ∈V, u ≠ v}

A tree is a connected graph with no cycles.

à ∃ a path between each pair of vertices.

4
 What is a Tree
• Abstract model of a
Computers”R”Us
hierarchical
structure
• A tree consists of nodes Sales Manufacturing R&D
with a parent-child
relation
• Applications: US International Laptops Desktops
– Organization charts
– File systems
– Programming Europe Asia Canada
environments

5
 Example: Genealogical Tree

CHARLES I

PHILIP II MARIA
JOAN

WENZEL
PHILIP III

ERNEST ALBERT
CHARLES 2 daugthers
MAXIMILIAN

MATHIAS
RUDOLPH II

Hasburg Family
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 Tree Terminology
• Root: node without parent (A) • Subtree: tree consisting
of a node and its
descendants
•Internal node: node with at least
one child (A, B, C, F) A

•External node (a.k.a. leaf ):

node without children (E, I, J, K, G, H, D) B C D

E F G H
•Ancestors of a node: parent, subtree
grandparent, grand-grandparent, etc.
I J K

•Descendant of a node: child, grandchild, grand-grandchild, etc. 7

 Tree Terminology

Distance between two nodes: number

of “edges” between them
A
•Depth of a node: number of
ancestors (= distance from the root)
•Height of a tree: maximum depth of B C D
any node (3)

E F G H

I J K

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 ADTs for Trees
• generic container methods
- size(), isEmpty(), elements()

• positional container methods

- positions(), swapElements(p,q), replaceElement(p,e)

• query methods
- isRoot(p), isInternal(p), isExternal(p)

• accessor methods
- root(), parent(p), children(p)

• update methods
- application specific

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Computing the depth of a node

If v is the root the depth is 0

If v is an internal node the depth is 1 + the depth of its parent

Algorithm depth(T,v)
if T.isRoot(v) then
return 0
else
return 1 + depth(T, T.parent(v))

Complexity ?
10
 Now, Traversing a tree!

How to visit all the nodes in a tree?

A

B C D

E F G H

I J K

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 Traversing Trees
Preorder Traversal

• A traversal visits the nodes of a Algorithm preOrder(v)

tree in a systematic manner visit(v)
• In a preorder traversal, a node is for each child w of v
visited before its descendants preorder (w)
• Application: print a structured
document

1
Make Money Fast!

2 5 9
1. Motivations 2. Methods References

6 7 8
3 4
2.1 Stock 2.2 Ponzi 2.3 Bank
1.1 Greed 1.2 Avidity
Fraud Scheme Robbery
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 Traversing Trees
Preorder Traversal

Algorithm preOrder(v)
visit(v)
for each child w of v
preorder (w)

D B A C F E H L I G
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 Traversing Trees
Postorder Traversal
• In a postorder traversal, a node Algorithm postOrder(v)
is visited after its for each child w of v
descendants postOrder (w)
• Application: compute space used visit(v)
by files in a directory and its
subdirectories
9
cs16/
8
3 7
todo.txt
homeworks/ programs/
1K

1 2 4 5 6
h1c.doc h1nc.doc DDR.java Stocks.java Robot.java
3K 2K 10K 25K 20K
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 Traversing Trees
Postorder Traversal
Algorithm postOrder(v)
for each child w of v do
recursively perform postOrder(w)
“visit” node v

A C B F L H I G E D

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 Traversing Trees
Inorder Traversal of a tree (Depth-first)

Let d(x) be the number

of sub-trees of node x.
Start: x = root
IN-ORDER VISIT
1. Visit the first sub-tree (inorder) A B C D FL H E I G

2. Visit the root

3. Visit the second sub-tree (inorder)
! !
d(x)+1. Visit the d(x)th sub-tree (inorder)

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17
 CHARLES I

PHILIP II

WENZEL
PHILIP III
ERNEST ALBERT
CHARLES MAXIMILIA
N
RUDOLPH II MATHIAS

When Charles dies, Philip II becomes King.

Charles I, If Philip II dies as well ….
Philip II,

Philip III, Charles,

Rudolph II, Ernest, Mathias, Max, Albert, Wenzel,

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 Binary Trees

right child
left child

Children are ordered

Each node has at most two children:
[0, 1, or 2] 19
“Full” Binary Trees (or “Proper”)

{
is a leaf, or
Each node:
has two children

Perfect Binary Trees

Full binary trees with all leaves at the same level:

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Complete Binary Trees

of depth h = Perfect trees of depth (h-1)

+
one or more leaves at level h.

Leaves go at the left

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 Examples of Binary Trees

Decision Tree
• Binary tree associated with a decision process
– internal nodes: questions with yes/no answer
– external nodes: decisions
• Example: dining decision

Want a fast meal?

Yes No

How about coffee? On expense account?

Yes No Yes No
Starbucks Spike’s Al Forno Café Paragon

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 Examples of Binary Trees
Arithmetic Expression Tree
• Binary tree associated with an arithmetic
expression
– internal nodes: operators
– external nodes: operands

Example: arithmetic expression

+
tree for the expression
× ×
(2 × (a - 1) + (3 × b))
2 − 3 b

a 1

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 Properties of Binary Trees
• Notation
n # of nodes e # of leaves
i # of internal nodes h height
Maximum number of
nodes at each level ?
Level 0 1

Level 1 2

Level 2
4

level i ------- 2i 24
 Properties of Full Binary Trees
• Notation • Properties:
n number of nodes – e=i+1
e number of leaves – n = 2e - 1
i number of internal nodes – h≤i
h height – h ≤ (n - 1)/2
– e ≤ 2h
– h ≥ log2 e
– h ≥ log2 (n + 1) - 1

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e=i+1

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e=i+1

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e=i+1

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n = 2e - 1

n=i+ e

e = i + 1 (just proved)

i = e -1

n = 2e - 1

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 h≤i
(h = max num of ancestors)

There must be at least one internal node for each level

(except the last) !

Ex: h=3, i=7 Ex: h=3, i=3

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e ≤ 2h level i ------- max num of nodes is 2i

h=3

23 leaves
if all at last
level h

otherwise less

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Since e ≤ 2h

log2 e ≤ log2 2h

log2 e ≤h

h ≥ log2 e

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 In Perfect Binary Trees…
with height h there are 2h+1 -1 nodes

l=0
n = 2h+1 -1
l=1

l=2

l=3

At each level there are 2l nodes, so the tree has:

h
∑ 2l = 1+ 2 + 4 + … + 2h = 2h+1-1
l=0

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As a consequence:

In Binary trees:
obviously n ≤ 2h+1 -1

n ≤ 2h+1-1

n+1 ≤ 2h+1

log (n+1) ≤ h+1

h ≥ log (n+1) -1

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 In Complete Binary Trees …

with height h 2h ≤ n ≤ 2h+1 - 1

2h- 1

From previous observation: n ≤ 2h+1 - 1

A complete binary tree is a perfect binary tree of height

h-1 plus some more leaves …

n ≥ 2h
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 n ≥ 2h

It follows that:

Height of a complete binary tree with

n nodes:

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 ADTs for Binary Trees

• accessor methods
-leftChild(p), rightChild(p), sibling(p)

• update methods
-expandExternal(p), removeAboveExternal(p)

other application specific methods

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 Traversing Binary Trees

Pre-, post-, in- (order)

• Refer to the place of the parent

relative to the children
• pre is before: parent, child, child
• post is after: child, child, parent
• in is in between: child, parent, child

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 Traversing Binary Trees
Preorder, Postorder,

Algorithm preOrder(T,v)
visit(v)
if v is internal:
preOrder (T,T.LeftChild(v))
preOrder (T,T.RightChild(v))

Algorithm postOrder(T,v)
if v is internal:
postOrder (T,T.LeftChild(v))
postOrder(T,T.RightChild(v))
visit(v)

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 Traversing Binary Trees
Inorder
(Depth-first)

Algorithm inOrder(T,v)
if v is internal:
inOrder (T,T.LeftChild(v))
visit(v)
if v is internal:
inOrder(T,T.RightChild(v))

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 Arithmetic Expressions

Inorder: a–b
−
Postorder: a b –
Preorder – a b
a b

Inorder:

2×a−1+3×b
+

× ×
Postorder:
2 − 3 b
2 a 1-× 3 b×+

a 1

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 a + (b • c – d)/e

PRE-ORDER:
+a/–•bcde

POST-ORDER:
abc•d–e/+

IN-ORDER:
a+b•c–d/e
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 Evaluate Arithmetic Expressions
• Specialization of a Algorithm evalExpr(v)
postorder traversal if isExternal (v)
– recursive method returning return v.element ()
the value of a subtree
– when visiting an internal else
node, combine the values x ← evalExpr(leftChild (v))
of the subtrees y ← evalExpr(rightChild (v))
◊ ← operator stored at v
return x ◊ y
+

× ×

2 − 3 2

5 1
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 +
× ×
2 − 3 2

5 1

Eval
Eval
×
×
2 − + 3 2
5 1

Eval Eval
Eval Eval
× − ×
3 2
2 5 1

−
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5 1
 Print Arithmetic Expressions
• Specialization of an inorder
Algorithm printExpression(v)
traversal if isInternal (v)
– print operand or operator when print(“(’’)
visiting node
– print “(“ before traversing left
inOrder (leftChild (v))
subtree print(v.element ())
– print “)“ after traversing right if isInternal (v)
subtree
inOrder (rightChild (v))
print (“)’’)

× ×
2× a−1 + 3×b
2 − 3 b
((2 × (a − 1)) + (3 × b))

a 1 45
Algorithm preOrderTraversalwithStack(T)
Stack S
TreeNode N

S.push(T) // push the reference to T in the empty stack

While (not S.empty())
N = S.pop()
if (N != null) {
print(N.elem) // print information
S.push(N.rightChild) // push the reference to
the right child
S.push(N.leftChild) // push the reference to
the left child
}

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 Algorithm preOrderTraversalwithStack(T)

S.push(T) // push the reference to T in the empty stack

N = S.pop()

print(N.elem)

b g

c d h i

e f

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 Algorithm preOrderTraversalwithStack(T)

S.push(T) // push the reference to T in the empty stack

N = S.pop()

print(N.elem)

N
T

b g

c d h i

e f
a

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 Algorithm preOrderTraversalwithStack(T)

S.push(N.rightChild) // push the reference to

the right child

S.push(N.leftChild) // push the reference to

the left child

b g

c d h i

e f
a
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 Algorithm preOrderTraversalwithStack(T)

N = S.pop()

b g

c d h i

e f
a
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 Algorithm preOrderTraversalwithStack(T)

N = S.pop()

print(N.elem)

N
T

b g

c d h i

e f
a b
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 Algorithm preOrderTraversalwithStack(T)

S.push(N.rightChild)

S.push(N.leftChild)

b g

c d h i

e f
a b
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 Euler Tour Traversal
• generic traversal of a binary tree
• the preorder, inorder, and postorder traversals are special
cases of the Euler tour traversal
• “walk around” the tree and visit each node three times:
– on the left
– from below
– on the right

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Algorithm eulerTour(T,v)

visit v (from the left)

if v is internal:
eulerTour (T,T.LeftChild(v))
visit v (from below)
if v is internal:
eulerTour(T,T.RightChild(v))
visit v (from the right)

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Implementations of Binary trees….

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 Implementing Binary Trees
with a Linked Structure

• A node is represented
by an object storing ∅
– Element
– Parent node
– Left child node
– Right child node B
• Node objects implement
the Position ADT
∅ ∅

B A D

A D ∅ ∅ ∅ ∅

C E C E
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 leftChild(p), rightChild(p), sibling(p):

Input: Position Output: Position

swapElements(p,q) Input: 2 Positions Output: None

replaceElement(p,e) Input: Position and an object Output: Object
isRoot(p) Input: Position Output: Boolean
isInternal(p) Input: Position Output: Boolean
isExternal(p) Input: Position Output: Boolean

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BTNode
Object Element
BTNode left, right, parent

Element
leftChild(v) return v.left

rightChild(v) return v.right

sibling(v)
p ← parent(v)
q ← leftChild(p)
if (v = q) return rightChild(p)
else return q

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replaceElement(v,obj)
temp ← v.element
v.element ← obj
return temp

swapElements(v,w)
temp ← w.element
w.element ← v.element
v.element ← temp

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leftChild(p), rightChild(p), sibling(p):

swapElements(p,q),
replaceElement(p,e)
isRoot(p), isInternal(p),
isExternal(p)

They all have

complexity O(1)

60
 Other interesting methods for the
ADT Binary Tree:
expandExternal(v): Transform v from an external
node into an internal node by creating two new children

B B
A D A D

C E C E

new1 new2

expandExternal(v):
new1 and new 2 are the new nodes
if isExternal(v)
v.left ← new1
v.right ← new2
size ← size +2

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removeAboveExternal(v):

B
B
A D
A D
C
C E
G
F G

A D

C
G

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 g p B
B
A D
A D s
C
C E

F G
v G

removeAboveExternal(v):
p if isExternal(v)
{ p ← parent(v)
s ← sibling(v)
s if isRoot(p) s.parent ← null and root ← s
B else
v A { g ← parent(p)
G
if p is leftChild(g) g.left ← s
else g.right ← s
s.parent ← g
}
size ← size -2 }

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Implementing Complete Binary Trees
with Vectors (Array-based)

1 2 3 4 5 6 7 8 9 10 11 12 13

H D I B E L O A C F G H N

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 leftChild(p), rightChild(p), sibling(p):

swapElements(p,q),
n = 11
replaceElement(p,e)
isRoot(p), isInternal(p),
isExternal(p)

They all have

complexity O(1)
Left child of T[i] T[2i] if 2i ≤ n

Right child of T[i] T[2i+1] if 2i + 1 ≤ n

Parent of T[i] T[i div 2] if i>1

The Root T[1] if T≠0
Leaf? T[i] TRUE if 2i > n
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leftChild(p), rightChild(p), sibling(p):

swapElements(p,q),
replaceElement(p,e)
isRoot(p), isInternal(p),
isExternal(p)

They all have

complexity O(1)

66
 Implementing General Trees
with a Linked Structure

• A node is represented by an
object storing
– Element
∅
– Parent node
– Sequence of children
nodes B
• Node objects implement
the Position ADT ∅ ∅

A D F
B

A D F

∅ ∅
C E
C E 67
 Representing General Trees
tree T

binary tree T' representing T

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 RULES

u in T u’ in T’

first child of u in T is left child of u’ in T’

first sibling of u in T is right child of u’ in T’

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 B
T

A D I

B
C E G L
H
stupid

T’
A
F
D

C
I

E
H
Place holder
G
L
F

70
 children are “completed” with “fake” nodes

The green squared nodes are the dummy nodes.

In this way ALL the original nodes are internal.

The leaves are the fake green nodes.
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 RULE:
to u in T corresponds u’ in T’

if u is a leaf in T and has no siblings, L L

then the children of u’ are leaves

If u is internal in T and v is its first child

then v’ is the left child of u’ in T’

D
D
C

C E G
D

If v has a sibling w immediately following it, C

w’ is the right child of v’ in T’
E 72
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