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CHAPTER 5 Trees

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Title: CHAPTER 5 Trees


1
CHAPTER 5Trees
All the programs in this file are selected
from Ellis Horowitz, Sartaj Sahni, and Susan
Anderson-Freed Fundamentals of Data Structures
in C, Computer Science Press,
1992. Fundamentals of Data Structures in C
2/E, 2007
2
Trees
Root
leaf
P.192 Fig 5.1 (2/E)
3
Definition of Tree
  • A tree is a finite set of one or more nodes such
    that
  • There is a specially designated node called the
    root.
  • The remaining nodes are partitioned into ngt0
    disjoint sets T1, ..., Tn, where each of these
    sets is a tree.
  • We call T1, ..., Tn the subtrees of the root.

4
Level and Depth
Level 1 2 3 4
Node (13) Degree of a node Leaf
(terminal) Nonterminal Parent Children Sibling Deg
ree of a tree (3) Ancestor Level of a node Height
of a tree (4)
Level
Degree
3
1
2
1
3
2
2
2
2
3
0
3
3
0
0
1
0
3
3
3
0
0
4
4
0
4
5
Terminology
  • The degree of a node is the number of subtreesof
    the node
  • The degree of A is 3 the degree of C is 1.
  • The node with degree 0 is a leaf or terminal
    node.
  • A node that has subtrees is the parent of the
    roots of the subtrees.
  • The roots of these subtrees are the children of
    the node.
  • Children of the same parent are siblings.
  • The ancestors of a node are all the nodes along
    the path from the root to the node.

6
Representation of Trees
  • List Representation
  • ( A ( B ( E ( K, L ), F ), C ( G ), D ( H ( M ),
    I, J ) ) )
  • The root comes first, followed by a list of
    sub-trees

data
link 1
link 2
...
link n
How many link fields are needed in such a
representation?
7
Left Child - Right Sibling
P.196 Fig 5.6 (2/E)
8
Binary Trees
  • A binary tree is a finite set of nodes that is
    either empty or consists of a root and two
    disjoint binary trees called the left subtree
    and the right subtree.
  • Any tree can be transformed into binary tree.
  • by left child-right sibling representation
  • The left subtree and the right subtree are
    distinguished.

9
Figure 5.6 Left child-right child tree
representation of a tree (p.191)
P. 197 Fig 5.7 (2/E)
A
B
C
E
D
G
F
K
H
L
I
M
J
10
Abstract Data Type Binary_Tree
structure Binary_Tree(abbreviated BinTree)
is objects a finite set of nodes either empty or
consisting of a root node, left Binary_Tree,
and right Binary_Tree. functions for all bt,
bt1, bt2 ? BinTree, item ? element Bintree
Create() creates an empty binary tree
Boolean IsEmpty(bt) if (btempty binary
tree) return TRUE else return FALSE
11
BinTree MakeBT(bt1, item, bt2) return a binary
tree whose left subtree is bt1, whose
right subtree is bt2, and whose root node
contains the data item Bintree Lchild(bt) if
(IsEmpty(bt)) return error
else return the left subtree of bt element
Data(bt) if (IsEmpty(bt)) return error
else return the data in the
root node of bt Bintree Rchild(bt) if
(IsEmpty(bt)) return error
else return the right subtree of bt
12
Samples of Trees
Complete Binary Tree
Skewed Binary Tree
1
2
3
4
5
P.200 Fig 5.10 (2/E)
13
Maximum Number of Nodes in BT
  • The maximum number of nodes on level i of a
    binary tree is 2i-1, igt1.
  • The maximum nubmer of nodes in a binary tree of
    depth k is 2k-1, kgt1.

Prove by induction.
14
Relations between Number ofLeaf Nodes and Nodes
of Degree 2
For any nonempty binary tree, T, if n0 is the
number of leaf nodes and n2 the number of nodes
of degree 2, then n0n21 proof Let n and
B denote the total number of nodes branches
in T. Let n0, n1, n2 represent the nodes with
no children, single child, and two children
respectively. n n0n1n2, B1n, Bn12n2
gt n12n21 n, n12n21 n0n1n2 gt
n0n21
15
Full BT VS Complete BT
  • A full binary tree of depth k is a binary tree of
    depth k having 2 -1 nodes, kgt0.
  • A binary tree with n nodes and depth k is
    complete iff its nodes correspond to the nodes
    numbered from 1 to n in the full binary tree of
    depth k.

k
????, ??????
1
1
2
2
3
3
5
7
6
7
4
6
4
5
13
14
12
15
11
10
9
8
8
9
Complete binary tree
Full binary tree of depth 4
16
Binary Tree Representations
  • If a complete binary tree with n nodes (depth
    log n 1) is represented sequentially, then
    forany node with index i, 1ltiltn, we have
  • parent(i) is at i/2 if i!1. If i1, i is at the
    root and has no parent.
  • left_child(i) ia at 2i if 2iltn. If 2igtn, then i
    has noleft child.
  • right_child(i) ia at 2i1 if 2i 1 ltn. If 2i 1
    gtn, then i has no right child.

17
Sequential Representation
1 2 3 4 5 6 7 8 9
A B C D E F G H I
(1) waste space (2) insertion/deletion
problem
A B -- C -- -- -- D -- . E
1 2 3 4 5 6 7 8 9 . 16
18
Linked Representation
typedef struct node tree_pointer typedef struct
node int data tree_pointer left_child,
right_child
data
right_child
data
left_child
right_child
left_child
19
Binary Tree Traversals
  • Let L, V, and R stand for moving left, visiting
    the node, and moving right.
  • There are six possible combinations of traversal
  • LVR, LRV, VLR, VRL, RVL, RLV
  • Adopt convention that we traverse left before
    right, only 3 traversals remain
  • LVR, LRV, VLR
  • inorder, postorder, preorder

20
Arithmetic Expression Using BT
inorder traversal A / B C D E infix
expression preorder traversal / A B C D
E prefix expression postorder traversal A B / C
D E postfix expression level order
traversal E D / C A B
P.206 Fig 5.16 (2/E)
21
Inorder Traversal (recursive version)
void inorder(tree_pointer ptr) / inorder tree
traversal / if (ptr)
inorder(ptr-gtleft_child) printf(d,
ptr-gtdata) inorder(ptr-gtright_child)

A / B C D E
P.207 Program 5.1 (2/E)
22
Preorder Traversal (recursive version)
void preorder(tree_pointer ptr) / preorder tree
traversal / if (ptr)
printf(d, ptr-gtdata)
preorder(ptr-gtleft_child)
preorder(ptr-gtright_child)
/ A B C D E
P.208 Program 5.2 (2/E)
23
Postorder Traversal (recursive version)
void postorder(tree_pointer ptr) / postorder
tree traversal / if (ptr)
postorder(ptr-gtleft_child)
postorder(ptr-gtright_child) printf(d,
ptr-gtdata)
A B / C D E
P.209 Program 5.3 (2/E)
24
Iterative Inorder Traversal (using stack)
void iterInorder(treepointer node) int top
-1 / initialize stack / treePointer
stackMAX_STACK_SIZE for () for (
node nodenode-gtleftChild) push(node)/
add to stack / nodepop() / delete from
stack / if (!node) break / empty stack /
printf(d, node-gtdata) node
node-gtrightChild
O(n)
P.210 Program 5.4 (2/E)
25
Trace Operations of Inorder Traversal
26
Level Order Traversal (using queue)
void levelOrder(treePointer ptr) / level order
tree traversal / int front rear 0
treePointer queueMAX_QUEUE_SIZE if (!ptr)
return / empty queue / addq(ptr) for ()
ptr deleteq()
27
if (ptr) printf(d, ptr-gtdata)
if (ptr-gtleftChild) addq(ptr-gtleftChild)
if (ptr-gtrightChild)
addq(ptr-gtrightChild) else break
E D / C A B
P.211 Program 5.5 (2/E)
28
Copying Binary Trees
treePointer copy(treePointer original) treePoint
er temp if (original) MALLOC(temp,
sizeof(temp)) temp-gtleftChildcopy(original-gtle
ftChild) temp-gtrightChildcopy(original-gtrightCh
ild) temp-gtdataoriginal-gtdata return
temp return NULL
postorder
P.212 Program 5.6 (2/E)
29
Equality of Binary Trees
the same topology and data
int equal(treePointer first, treePointer
second) / function returns FALSE if the binary
trees first and second are not equal,
otherwise it returns TRUE / return ((!first
!second) (first second
(first-gtdata second-gtdata)
equal(first-gtleftChild, second-gtleftChild)
equal(first-gtrightChild, second-gtrightChild)))

P.213 Program 5.7 (2/E)
30
Threaded Binary Trees
  • Two many null pointers in current
    representationof binary trees n number of
    nodes number of non-null links n-1 total
    links 2n null links 2n-(n-1)n1
  • Replace these null pointers with some useful
    threads.

31
Threaded Binary Trees (Continued)
If ptr-gtleftChild is null, replace it with a
pointer to the node that would be visited
before ptr in an inorder traversal If
ptr-gtrightChild is null, replace it with a
pointer to the node that would be visited
after ptr in an inorder traversal
32
A Threaded Binary Tree
root
dangling
dangling
inorder traversal H, D, I, B, E, A, F, C, G
P.218 Fig 5.21 (2/E)
33
Data Structures for Threaded BT
left_thread left_child data
right_child right_thread
?
TRUE
?
FALSE child
TRUE thread
typedef struct threaded_tree threaded_pointer ty
pedef struct threaded_tree short int
left_thread threaded_pointer left_child
char data threaded_pointer right_child
short int right_thread
34
Memory Representation of A Threaded BT
root
--
f
f
A
f
f
C
B
f
f
f
f
G
E
F
t
t
D
t
t
t
f
t
f
I
H
t
t
t
t
P.219 Fig 5.23 (2/E)
35
Next Node in Threaded BT
threaded_pointer insucc(threaded_pointer tree)
threaded_pointer temp temp
tree-gtright_child if (!tree-gtright_thread)
while (!temp-gtleft_thread) temp
temp-gtleft_child return temp
36
Inorder Traversal of Threaded BT
void tinorder(threadedPointer tree) / traverse
the threaded binary tree inorder /
threadedPointer temp tree for ()
temp insucc(temp) if (temptree)
break printf(3c, temp-gtdata)
O(n)
P.221 Program 5.11 (2/E)
37
Inserting Nodes into Threaded BTs
  • Insert child as the right child of node parent
  • change parent-gtright_thread to FALSE
  • set child-gtleft_thread and
  • child-gtright_thread to TRUE
  • set child-gtleft_child to point to parent
  • set child-gtright_child to
  • parent-gtright_child
  • change parent-gtright_child to point to child

38
Examples
Insert a node D as a right child of B.
root
root
A
A
(1)
parent
B
parent
B
(3)
child
child
C
C
D
D
(2)
empty
39
(3)
(1)
(2)
(4)
nonempty
Figure 5.24 Insertion of child as a right child
of parent in a threaded binary tree (p.222)
40
Right Insertion in Threaded BTs
void insertRight(threadedPointer parent,
threadedPointer child)
threadedPointer temp child-gtrightChild
parent-gtrightChild child-gtrightThread
parent-gtrightThread child-gtleftChild parent
case (a) child-gtleftThread TRUE
parent-gtrightChild child parent-gtrightThread
FALSE if (!child-gtrightThread) case (b)
temp insucc(child) temp-gtleftChild
child
(1)
(2)
(3)
(4)
41
Heap
  • A max tree is a tree in which the key value in
    each node is no smaller than the key values in
    its children. A max heap is a complete binary
    tree that is also a max tree.
  • A min tree is a tree in which the key value in
    each node is no larger than the key values in
    its children. A min heap is a complete binary
    tree that is also a min tree.
  • Operations on heaps
  • creation of an empty heap
  • insertion of a new element into the heap
  • deletion of the largest element from the heap

42
Figure 5.25 Sample max heaps (1/E p.219)
P.225 Fig 5.25 (2/E)
4
Property The root of max heap (min heap)
contains the largest (smallest).
43
Figure 5.26Sample min heaps (1/E p.220)
P.225 Fig 5.26 (2/E)
44
ADT for Max Heap
structure MaxHeap objects a complete binary
tree of n gt 0 elements organized so that the
value in each node is at least as large as those
in its children functions for all heap
belong to MaxHeap, item belong to Element, n,
max_size belong to integer MaxHeap
Create(max_size) create an empty heap that can
hold a
maximum of max_size elements Boolean
HeapFull(heap, n) if (nmax_size) return
TRUE
else return FALSE MaxHeap Insert(heap,
item, n) if (!HeapFull(heap,n)) insert
item into heap and
return the resulting heap else return
error Boolean HeapEmpty(heap, n) if (ngt0)
return FALSE
else return TRUE Element
Delete(heap,n) if (!HeapEmpty(heap,n)) return
one instance of
the largest element in the heap
and remove it from the heap
else
return error
45
Application priority queue
  • machine service
  • amount of time (min heap)
  • amount of payment (max heap)
  • factory
  • time tag

46
Data Structures
  • unordered linked list
  • unordered array
  • sorted linked list
  • sorted array
  • heap

47
Figure 5.27 Priority queue representations (1/E
p.221)
48
Example of Insertion to Max Heap
21
20
20
20
15
5
2
15
15
2
14
10
2
10
14
14
10
insert 21 into heap
insert 5 into heap
initial location of new node
49
Insertion into a Max Heap (2/E)
void push (element item, int n) / insert item
into a max heap of current size n / int i
if (HEAP_FULL(n)) fprintf(stderr, The
heap is full.\n) exit(EXIT_FAILURE)
i(n) while ((i!1) (item.keygtheapi/2.k
ey)) heapiheapi/2 i/2
heapiitem
P.227 Program 5.13 (2/E)
2k-1n gt k?log2(n1)?
O(log2n)
50
Insertion into a Max Heap (1/E)
void insert_max_heap(element item, int n)
int i if (HEAP_FULL(n)) fprintf(stderr,
the heap is full.\n) exit(1) i
(n) while ((i!1)(item.keygtheapi/2.key))
heapi heapi/2 i / 2
heapi item
2k-1n gt k?log2(n1)?
O(log2n)
51
Example of Deletion from Max Heap
remove
10
15
20
14
2
2
15
2
15
14
10
10
14
(a) Heap structure
(c) Finial heap
(b) 10 inserted at the root
52
Deletion from a Max Heap (2/E)
P.229 Program 5.14 (2/E)
element pop(int n) / delete element with the
highest key from the heap/ int parent, child
element item, temp if (HEAP_EMPTY(n))
fprintf(stderr, The heap is empty\n)
exit(EXIT_FAILURE) / save value of the
element with the highest key / item
heap1 / use last element in heap to adjust
heap / temp heap(n)-- parent 1
child 2
53
while (child lt n) / find the larger
child of the current parent / if
((child lt n) (heapchild.keyltheapchil
d1.key)) child if (temp.key gt
heapchild.key) break / move to the next
lower level / heapparent heapchild
parent child child 2
heapparent temp return item
54
Deletion from a Max Heap (1/E)
element delete_max_heap(int n) int parent,
child element item, temp if
(HEAP_EMPTY(n)) fprintf(stderr, The heap
is empty\n) exit(1) / save value of
the element with the highest key / item
heap1 / use last element in heap to adjust
heap / temp heap(n)-- parent 1
child 2
55
while (child lt n) / find the larger
child of the current parent / if
((child lt n) (heapchild.keyltheapchil
d1.key)) child if (temp.key gt
heapchild.key) break / move to the next
lower level / heapparent heapchild
child 2 heapparent temp return
item
56
Binary Search Tree
  • Heap
  • a min (max) element is deleted. O(log2n)
  • deletion of an arbitrary element O(log2n)
  • search for an arbitrary element O(n)
  • Binary search tree
  • Every element has a unique key.
  • The keys in a nonempty left subtree (right
    subtree) are smaller (larger) than the key in the
    root of subtree.
  • The left and right subtrees are also binary
    search trees.

57
Examples of Binary Search Trees
60
30
20
70
40
25
15
5
80
65
12
2
22
(c)
(b)
10
(a)
58
Searching a Binary Search Tree (recursive)
P.233 Program 5.15 (2/E)
element search(treePointer root, int
k) / ???parameter???????(by TA) / / return a
pointer to the node that contains key. If there
is no such node, return NULL / if (!root)
return NULL if (k root-gtdata.key)
return (root-gtdata) if (key lt root-gtdata.key)
return search(root-gtleftChild,
k) return search(root-gtrightChild,key)

59
Another Searching Algorithm (iterative)
P.233 Program 5.16 (2/E)
element iterSearch(treePointer tree, int k) /
return a pointer to the element whose key is k,
if theres no such element, return NULL, /
while (tree) if (key tree-gtdata.key)
return (tree-gtdata) if (key lt
tree-gtdata.key) tree tree-gtleftChild
else tree tree-gtrightChild return
NULL
O(h)
60
Insert Node in Binary Search Tree
30
30
30
40
5
40
40
5
5
80
35
2
80
2
2
Insert 35
Insert 80
61
Insertion into A dictionary pair into a Binary
Search Tree
void insert(treePointer node, int k, iType
theItem) / if k is in the tree pointed at by
node do nothing otherwise add a new node with
data (k, theItem) / treePointer ptr
treePointer temp modifiedSearch(node, k) if
(temp !(node)) / k isnt in the tree /
MALLOC(ptr, sizeof(ptr)) ptr-gtdata.key
k ptr-gtleftChild ptr-gtrightChild NULL
if (node) / insert as child of temp /
if (k lt temp-gtdata) temp-gtleftChildptr
else temp-gtrightChild ptr else node
ptr
P.235 Program 5.17 (2/E)
62
Deletion for A Binary Search Tree
1
leaf node
30
80
5
T2
T1
1
2
2
X
T1
T2
63
Deletion for A Binary Search Tree
non-leaf node
40
40
55
60
20
20
70
30
50
70
10
30
50
10
52
45
55
45
52
After deleting 60
Before deleting 60
64
1
2
T1
T3
T2
1
2
T1
T3
T2
65
Forest
  • A forest is a set of n gt 0 disjoint trees

A
Forest
G
E
A
B
E
I
H
F
G
F
D
C
C
B
H
D
I
66
Transform a forest into a binary tree
  • T1, T2, , Tn a forest of treesB(T1, T2, ,
    Tn) a binary tree corresponding to this forest
  • algorithm(1) empty, if n 0(2) has root equal
    to root(T1) has left subtree equal to
    B(T11,T12,,T1m) has right subtree equal to
    B(T2,T3,,Tn)

67
Forest Traversals
  • Preorder
  • If F is empty, then return
  • Visit the root of the first tree of F
  • Taverse the subtrees of the first tree in tree
    preorder
  • Traverse the remaining trees of F in preorder
  • Inorder
  • If F is empty, then return
  • Traverse the subtrees of the first tree in tree
    inorder
  • Visit the root of the first tree
  • Traverse the remaining trees of F is indorer

68
inorder EFBGCHIJDA preorder ABEFCGDHIJ
A
B
A
C
E
B
D
C
D
G
F
E
F
J
H
I
G
H
I
B E F
C G
preorder
D H I J
J
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