Post on 04-Jan-2016
CHAPTER 06
Compiled by: Dr. Mohammad Omar AlhawaratTrees
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Definition of Tree
A tree is a set of linked nodes, such that there is one and only one path from a unique node (called the root node) to every other node in the tree
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Hierarchical Organization
Example: File Directories
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Tree Terminology
Nodes at a given level are children of nodes of previous level
Node with children is the parent node of those children
Nodes with same parent are siblings Node with no children is a leaf node The only node with no parent is the root
node All others have one parent each
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Example of a Tree
A
C B D
E F G
root
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Example of a Tree
A
C B D
E F G
rootIn a tree, every pair of linked nodes have a parent-child relationship (the parent is closer to the root)
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Example of a Tree (Cont.)
A
C B D
E F G
root For example, C is a parent of G
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Example of a Tree (Cont.)
A
C B D
E F G
root E and F are children of D
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Example of a Tree (Cont.)
A
C B D
E F G
root The root node is the only node that has no parent.
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Example of a Tree (Cont.)
A
C B D
E F G
root Leaf nodes (or leaves for short) have no children.
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Subtrees
A
B C
I K D E F
J G H
subtree
root A subtree is a part of a tree that is a tree in itself
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Subtrees (Cont.)
A
B C
I K D E F
J G H
subtree
root It normally includes each node reachable from its root.
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Subtrees (Cont.)
A
B C
I K D E F
J G H
root Even though this looks like a subtree in itself…
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Binary Trees
A binary tree is a tree in which each node can only have up to two children…
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Not a Binary Tree
A
B C
I K D E F
J G H
root
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Example of a Binary Tree
A
B C
I K E
J G H
rootThe links in a tree are often called edges
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Levels
A
B C
I K E
J G H
rootlevel 0
level 1
level 2
level 3
The level of a node is the number of edges in the path from the root node to this node
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Full Binary Tree
B
D
H
root
I
A
E
J K
C
F
L M
G
N O
In a full binary tree, each node has two children except for the nodes on the last level, which are leaf nodes
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Complete Binary Trees
A complete binary tree is a binary tree that is either a full binary tree OR a tree that would be a full binary tree but it
is missing the rightmost nodes on the last level
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Complete Binary Trees (Cont.)
B
D
H
root
I
A
E
C
F G
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Complete Binary Trees (Cont.)
B
D
H
root
I
A
E
J K
C
F
L
G
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Full Binary Trees
B
D
H
root
I
A
E
J K
C
F
L M
G
N O
A full binary tree is also a complete binary tree.
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The formula for the maximum number of nodes is derived from the fact that each node can have only two descendants. Given a height of the binary tree, H, the maximum number of nodes in the full binary tree is given as follows:
max 2 1HN
Full Binary Trees (Cont.)
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To Assure that a binary tree is Balanced one the following algorithms is used:
AVL Trees. Red-Black Trees. 2-3 Trees and the general case (B-Trees)
B-Trees are used with Fetching data from Large Databases.
Balanced Binary Trees
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Binary Trees
A binary tree is either empty or has the following form
Where Tleft and Tright are binary trees
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Binary Trees
Every nonleaf in a full binary tree has exactly two children
A complete binary tree is full to its next-to-last level Leaves on last level filled from left to right
The height of a binary tree with n nodes that is either complete or full is log2(n + 1)
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Binary Trees
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A tree is a set of nodes that either: is empty or has a designated node, called the root,
from which hierarchically descend zero or more subtrees, which are also trees.
Recursive definition of a tree
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Recursive definition of a tree (Cont.)
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Tree Traversal
Four meaningful orders in which to traverse a binary tree. Preorder Inorder Postorder Level order
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Tree Traversal (Cont.)
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In Preorder, the rootis visited before (pre)the subtrees traversals
In Inorder, the root isvisited in-between left and right subtree traversal
In Postorder, the rootis visited after (post)the subtrees traversals
Preorder Traversal:1. Visit the root2. Traverse left subtree3. Traverse right subtree
Inorder Traversal:1. Traverse left subtree2. Visit the root3. Traverse right subtree
Postorder Traversal:1. Traverse left subtree2. Traverse right subtree3. Visit the root
Tree Traversal (Cont.)
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Tree Traversal (Cont.)
Visiting a node Processing the data within a node
This is the action performed on each node during traversal of a tree
A traversal can pass through a node without visiting it at that moment
For a binary tree Visit the root Visit all nodes in the root's left subtree Visit all nodes in the root's right subtree
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Tree Traversal (Cont.)
Preorder traversal: visit root before the subtrees
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Tree Traversal (Cont.)
Inorder traversal: visit root between visiting the subtrees
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Tree Traversal (Cont.)
Postorder traversal: visit root after visiting the subtrees
These are examples of a depth-first traversal.
These are examples of a depth-first traversal.
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Tree Traversal (Cont.)
Level-order traversal: begin at the root, visit nodes one level at a time
This is an example of a breadth-first traversal.
This is an example of a breadth-first traversal.
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Tree Traversals – Example 1
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Assume: visiting a node is printing its label
Preorder: 1 3 5 4 6 7 8 9 10 11 12
Inorder:4 5 6 3 1 8 7 9 11 10 12
Postorder:4 6 5 3 8 11 12 10 9 7 1
1
3
11
98
4 6
5
7
12
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Tree Traversals – Example 2
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Assume: visiting a node is printing its data
Preorder: 15 8 2 6 3 711 10 12 14 20 27 22 30
Inorder: 2 3 6 7 8 10 1112 14 15 20 22 27 30
Postorder: 3 7 6 2 10 1412 11 8 22 30 27 20 15
6
15
8
2
3 7
11
10
14
12
20
27
22 30
Tree Traversals – Example 3
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Tree Traversals – Example 4
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Examples of Binary Trees
Expression Trees
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Tree Traversal - Summary
Level order traversal is sometimes called breadth-first.
The other traversals are called depth-first.
Traversal takes Θ(n) in both breadth-first and depth-first.
Memory usage in a perfect tree is Θ(log n) in depth-first and Θ(n) in breadth-first traversal.
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Questions?
Questions ?