Binary Trees
(and Big “O” notation)
CS-2303
System Programming Concepts
(Slides include materials from The C Programming Language, 2nd edition, by Kernighan and Ritchie and
from C: How to Program, 5th and 6th editions, by Deitel and Deitel)
CS-2303, C-Term 2010
Binary Trees
1
Definitions
• Linked List
• A data structure in which each element is
dynamically allocated and in which elements point
to each other to define a linear relationship
• Singly- or doubly-linked
• Stack, queue, circular list
• Tree
• A data structure in which each element is
dynamically allocated and in which each element
has more than one potential successor
• Defines a partial order
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Binary Trees
2
Binary Tree
• A linked list but with
two links per item
struct treeItem {
type payload;
treeItem *left;
treeItem *right;
};
payload
left
payload
left
payload
payload
left
left
right
payload
right
left
payload
left
right
right
right
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Binary Trees
3
right
payload
left
right
Binary Tree (continued)
• Binary tree needs a root
struct treeItem {
type payload;
treeItem *left; treeItem *right;
};
struct treeItem *root;
• Binary trees often drawn with root at top!
• Unlike ordinary trees in the forest
• More like the root systems of a tree
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Binary Trees
4
Definitions (continued)
See Deitel & Deitel, §12.7
K & R, §6.5
• Binary Tree
• A tree in which each element has two potential
successors
• Subtree
• The set of nodes that are successors to a specific
node, either directly or indirectly
• Root of a tree
• The node of the tree that is not the successor to any
other node, all other nodes are (directly or
indirectly) successors to it
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Binary Trees
5
Binary Tree
struct treeItem {
type payload;
treeItem *left;
treeItem *right;
payload
left
};
struct treeItem *root;
right
payload
payload
left
left
right
right
payload
left
payload
right
left
payload
left
right
right
CS-2303, C-Term 2010
Binary Trees
6
payload
left
right
Purpose of a Tree
• (Potentially) a very large data structure
• Capable of storing very many items
• In an orderly way
• Need to find items by value
• I.e., need to search through the data structure to see if it
contains an item with the value we want
• Need to add new items
• If value is not already in the tree, add a new item …
• …so that it can be easily found in future
• Why not use a linked list?
CS-2303, C-Term 2010
Binary Trees
7
Searching and Adding to a Binary Tree
• Look recursively down
sequence of branches until
either
payload
left
– Desired node is found; or
– Null branch is encountered
• Replace with ptr to new item
• Decide which branch to
follow based on payload
right
payload
payload
left
left
right
right
payload
left
payload
right
left
payload
left
right
right
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Binary Trees
8
payload
left
right
Example — Searching a Tree
typedef struct _treeItem {
char *word;
// part of payload
int count;
// part of payload
_treeItem *left, *right;
} treeItem;
treeItem *findItem(treeItem *p, char *w) {
if (p == NULL)
return NULL; // item not found
int c = strcmp(w, p->word);
if (c == 0)
return p;
else if (c < 0)
return findItem(p->left, w);
else
return findItem(p->right, w);
}
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Binary Trees
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Example — Adding an Item
treeItem *addItem(treeItem *p, char *w) {
if (p == NULL){
p = malloc(sizeof(treeItem));
char *c = malloc(strlen(w)+1);
Why do this?
p->word = strcpy(c, w);
p->count = 1;
p->left = p->right = NULL;
return p;
};
int c = strcmp(w, p->word);
if (c == 0)
p->count++;
else if (c < 0)
p->left = addItem(p->left, w);
else
p->right = addItem(p->right, w);
return p;
}
CS-2303, C-Term 2010
Binary Trees
10
Binary Tree
• Question:– how many
calls to addItem for
a tree with 106 nodes?
– Assume balanced
– I.e., approx same number of
nodes on each subtree
payload
left
right
payload
payload
left
left
right
right
payload
left
payload
right
left
payload
left
right
right
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Binary Trees
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payload
left
right
Answer
• Approximately 20 calls to addItem
• Note:–
– 210 = 1024  103
– Therefore 106  220
– Therefore it takes approximately 20 two-way branches
to cover 106 items!
• How many comparisons would it take to search a
linked list of 106 items?
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Binary Trees
12
Observation
• Problems like this occur in real life all the
time
• Need to maintain a lot of data
• Usually random
• Need to search through it quickly
• Need to add (or delete) items dynamically
• Need to sort “on the fly”
• I.e., as you are adding and/or deleting items
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Binary Trees
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Questions?
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Binary Trees
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Binary Trees (continued)
• Binary tree does not need to be “balanced”
• i.e., with approximate same # of nodes hanging from
right or left
• However, it often helps with performance
• Multiply-branched trees
• Like binary trees, but with more than two links per
node
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Binary Trees
15
Binary Trees (continued)
• Binary tree does not need to be “balanced”
• i.e., with approximate same # of nodes hanging from
right or left
• However, it helps with performance
• Time to reach a leaf node is O(log2 n), where n is
number of nodes in tree
• Multiply-branched trees
• Like binary trees, but with more than two links per
node
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Binary Trees
16
Order of Traversing Binary Trees
• In-order
• Traverse left sub-tree (in-order)
• Visit node itself
• Traverse right sub-tree (in-order)
• Pre-order
• Visit node first
• Traverse left sub-tree
• Traverse right sub-tree
• Post-order
• Traverse left sub-tree
• Traverse right sub-tree
• Visit node last
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Binary Trees
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Question
• Suppose we wish to print out the strings
stored in the tree of the previous example in
alphabetical order?
• What traversal order of the tree should we
use?
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Binary Trees
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Another Example of Binary Tree
x = (a.real*b.imag - b.real*a.imag) /
sqrt(a.real*b.real – a.imag*b.imag)
=
x
/
sqrt
*
.
a
-
*
.
real
CS-2303, C-Term 2010
b
.
imag
b
Binary Trees
…
.
real
a
19
imag
Question
• What kind of traversal order is required for
this expression?
• In-order?
• Pre-order?
• Post-order?
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Binary Trees
20
Binary Trees in Compilers
• Used to represent the structure of the
compiled program
Note: Deitel & Deitel, Ch 12 exercises,
• Optimizations contain a series on building a compiler
•
•
•
•
•
Common sub-expression detection
Code simplification
Loop unrolling
Parallelization
Reductions in strength – e.g., substituting additions
for multiplications, etc.
• Many others
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Binary Trees
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Questions?
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Binary Trees
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“Big O” notation
New Topic
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Binary Trees
23
Linked Lists Again
• Linear data structure
• Easy to grow and shrink
• Easy to add and delete items
• Time to search for an item – O(n)
CS-2303, C-Term 2010
Binary Trees
24
Binary Trees Again
• Non-linear data structure
• Easy to grow and shrink
• Easy to add and delete items
• Time to search for an item – O(log n)
CS-2303, C-Term 2010
Binary Trees
25
Definition — Big-O
“Of the order of …”
• A characterization of the number of
operations in an algorithm in terms of a
mathematical function of the number of
data items involved
• O(n) means that the number of operations to
complete the algorithm is proportional to n
• E.g., searching a list with n items requires,
on average, n/2 comparisons with payloads
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Binary Trees
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Big-O (continued)
•
•
•
•
•
O(n): proportional to n – i.e., linear
O(n2): proportional to n2 – i.e., quadratic
O(kn) – proportional to kn – i.e., exponential
…
O(log n) – proportional to log n – i.e.,
sublinear
• O(n log n)
• Worse than O(n), better than O(n2)
• O(1) – independent of n; i.e., constant
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Binary Trees
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Anecdote & Questions:–
• In the design of electronic adders, what is the
order of the carry-propagation?
• What is the order of floating point divide?
• What is the order of floating point square root?
• What program have we studied in this course that
is O(2n)? i.e., exponential?
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Binary Trees
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Questions on Big-O?
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Binary Trees
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