Transcript Powerpoint Slides for the Standard Version of Starting Out

Trees
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Outline
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Tree Structures
Tree Node Level and Path Length
Binary Tree Definition
Binary Tree Nodes
Binary Search Trees
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Tree Structures
Arrays, vectors, lists are linear structures, one element
follows another. Trees are hierarchical structure.
President -CEO
Product ion
Manager
Personnel
Manager
Purchasing
Supervisor
Warehouse
Supervisor
Sales
Manager
Shipping
Supervisor
HIERARC HIC AL TREE STRUC TURE
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Tree Structures
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BINARY EXPRES S ION TREE FOR
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"a*b + (c-d) / e "
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Tree Terminology
Tree is a set of nodes. The set may be empty. If not
empty, then there is a distinguished node r, called
root, all other nodes originating from it, and zero or
more non-empty subtrees T1,T2, …,Tk, each of whose
roots are connected by a directed edge from r.
(inductive definition)
r
T1
T2
…
Tk
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Tree Terminology
 root
A
Nodes in subtrees are called successors or
descendents of the root node
An immediate successor is called a child
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A leaf node is a node without any children while
an interior node has at least one child.
The link between node describes the parentchild relation.
The link between parent and child is called edge
(b)
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(a)
A path between a parent P and any node N in its
subtrees is a sequence of nodes P= X0,X1,
…,Xk=N, where k is the length of the path. Each
node Xi in the path is the parent of Xi+1, (0≤i<k)
 Size of tree is the number of nodes in the tree
A GENERAL TREE
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Tree Node Level and Path
Length
The level of a node N in a tree is the length of the path from root to N.
Root is at level 0.
The depth of a tree is the length of the longest path from root to any
node.
Level: 0
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B
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Level: 1
Level: 2
Level: 3
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Binary Tree
 In a binary tree, no node has more than two
children, and the children are distinguished as
left and right.
 A binary tree T is a finite set of nodes with one
the following properties:
1. T is a tree if the set of nodes is empty.
2. The set consists of a root R and exactly two
distinct binary trees. The left subtree TL and
the right subtree TR, either or both subtree
may be empty.
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Examples of Binary Trees
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Tree A
Size 9 Depth 3
Size? depth?
Tree B
Size 5 Depth 4
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Full Binary Tree
• full binary tree: a binary tree is which each node
was exactly 2 or 0 children
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Density of a Binary Tree
 At any level n, a binary tree may contain from
1 to 2n nodes. The number of nodes per level
contributes to the density of the tree.
 Degenerate tree: there is a single leaf node
and each interior node has only one child.
 An n-node degenerate tree has depth n-1
 Equivalent to a linked list
 A complete binary tree of depth n is a tree in
which each level from 0 to n-1 has all possible
nodes and all leaf nodes at level n occupy the
leftmost positions in the tree.
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Complete Binary Tree
• complete binary tree: a binary tree in which every
level, except possibly the deepest is completely
filled. At depth n, the height of the tree, all nodes
are as far left as possible
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Complete or noncomplete?
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Complete Tree (Depth 2)
Full with all possible nodes
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Complete or noncomplete?
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Non-Complete Tree (Depth 3)
Level 2 is missing nodes
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Complete or noncomplete?
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B
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Non-CompleteTree (Depth 3)
occup leftmost positions
Nodes at level 3 do not occurpy
y
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Complete or noncomplete?
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B
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Complete Tree (Depth 3)
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Perfect Binary Tree
• perfect binary tree: a binary tree with all leaf nodes
at the same depth. All internal nodes have exactly
two children.
• a perfect binary tree has the maximum number of
nodes for a given height
• a perfect binary tree has 2(n+1) - 1 nodes where n is
the height of a tree
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height = 0 -> 1 node
height = 1 -> 3 nodes
height = 2 -> 7 nodes
height = 3 -> 15 nodes
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Binary Search Trees
• Key property
– Value at node
• Smaller values in left subtree
• Larger values in right subtree
– Example
X
• X>Y
• X<Z
Y
Z
Binary Search Trees
• Examples
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10
2
5
10
45
30
5
45
30
2
25
45
2
10
25
30
25
Binary
search trees
Not a binary
search tree
Iterative Search of Binary Tree
Node *Find( Node *n, int key) {
while (n != NULL) {
if (n->data == key)
// Found it
return n;
if (n->data > key)
// In left subtree
n = n->left;
else
// In right subtree
n = n->right;
}
return null;
}
Node * n = Find( root, 5);
Recursive Search of Binary
Tree
Node *Find( Node *n, int key) {
if (n == NULL)
// Not found
return( n );
else if (n->data == key) // Found it
return( n );
else if (n->data > key)
// In left subtree
return Find( n->left, key );
else
// In right subtree
return Find( n->right, key );
}
Node * n = Find( root, 5);
Example Binary Searches
• Find ( root, 2 )
root
10
5
5
10 > 2, left
30
5 > 2, left
2
45
2 = 2, found
2
25
30
45
5 > 2, left
10
25
2 = 2, found
Example Binary Searches
• Find (root, 25 )
10
5
5
10 < 25, right
30
30 > 25, left
2
45
25 = 25, found
2
25
5 < 25, right
45 > 25, left
30 > 25, left
30
45
10
10 < 25, right
25 = 25, found
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Types of Binary Trees
• Degenerate – only one child
• Complete – always two children
• Balanced – “mostly” two children
– more formal definitions exist, above are intuitive ideas
Degenerate
binary tree
Balanced
binary tree
Complete
binary tree
Binary Search Tree
Construction
• How to build & maintain binary trees?
– Insertion
– Deletion
• Maintain key property (invariant)
– Smaller values in left subtree
– Larger values in right subtree
Binary Search Tree – Insertion
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Algorithm
1. Perform search for value X
2. Search will end at node Y (if X not in tree)
3. If X < Y, insert new leaf X as new left subtree for Y
4. If X > Y, insert new leaf X as new right subtree for Y
Observations
– O( log(n) ) operation for balanced tree
– Insertions may unbalance tree
Example Insertion
• Insert ( 20 )
10
5
10 < 20, right
30 > 20, left
30
25 > 20, left
2
25
20
45
Insert 20 on left
Binary Search Tree – Deletion
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Algorithm
1. Perform search for value X
2. If X is a leaf, delete X
3. Else // must delete internal node
a) Replace with largest value Y on left subtree
OR smallest value Z on right subtree
b) Delete replacement value (Y or Z) from subtree
Observation
– O( log(n) ) operation for balanced tree
– Deletions may unbalance tree
Example Deletion (Leaf)
• Delete ( 25 )
10
5
10
10 < 25, right
30
5
30 > 25, left
30
25 = 25, delete
2
25
45
2
45
Example Deletion (Internal
Node)
• Delete ( 10 )
10
5
2
5
30
25
5
45
Replacing 10
with largest
value in left
subtree
2
5
30
25
2
45
Replacing 5
with largest
value in left
subtree
2
30
25
45
Deleting leaf
Example Deletion (Internal
Node)
• Delete ( 10 )
10
5
2
25
30
25
5
45
Replacing 10
with smallest
value in right
subtree
2
25
30
25
5
45
Deleting leaf
2
30
45
Resulting tree