trees chapter 15. 2 chapter contents tree concepts hierarchical organizations tree terminology...
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TRANSCRIPT
2
Chapter Contents
Tree Concepts Hierarchical
Organizations Tree Terminology
Traversals of a Tree Traversals of a Binary
Tree Traversals of a General
Tree
Java Interfaces for Trees
Interfaces for All Trees Interface for Binary
Examples of Binary Trees Expression Trees Decision Trees Binary Search Trees
Examples of General Trees Parse Trees Game Trees
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Tree Concepts
Previous data organizations place data in linear order
Some data organizations require categorizing data into groups, subgroups
This is hierarchical classification Data items appear at various levels within the
organization
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Hierarchical Organization
Example: A university's organization
A university's administrative structure.
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Tree Terminology
A tree is A set of nodes Connected by edges
The edges indicate relationships among nodes
Nodes arranged in levels Indicate the nodes' hierarchy Top level is a single node called the root
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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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Tree Terminology
Empty trees? Some authors specify a general tree must have at least the
root node This text will allow all trees to be empty
A node is reached from the root by a path The length of the path is the number of edges that compose
it
The height of a tree is the number of levels in the tree The subtree of a node is a tree rooted at a child of
that node
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Binary Trees General tree: in general, node can have an arbitrary
number of children Binary tree if each node has at most two children
Three 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 If a binary tree of height h has all leaves on the same level h and
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
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Binary Trees
Total number of nodes n for a full tree can be calculated as:
The height of a binary tree with n nodes that is either complete or full is
log2(n + 1)
1
0
2 2 1h
i h
i
n
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Traversals of a Tree Traversing items is a common operation.
Previous chapters, data was arranged linearly. Order of traversal was clear.
During tree traversal, we must visit each item exactly once in a systematic way. The order is not unique.
Visiting a node Processing the data within a node
A traversal can pass through a node without visiting it at that moment
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Traversals of a Tree
Subtree of the root of binary trees are themselves binary trees. Using this recursive nature, we define recursive traversal:
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
Order does not matter. Visit root before, between or after visiting two subtrees.
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Traversals of a Tree
Preorder traversal: visit root before the subtrees
The visitation order of a preorder traversal.
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Traversals of a Tree
Inorder traversal: visit root between visiting the subtrees
The visitation order of an inorder traversal.
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Traversals of a Tree
Postorder traversal: visit root after visiting the subtrees
The visitation order of a postorder traversal.
These are examples of a depth-first traversal.
Follow a path that descends the levels of a
tree as deeply as possible until reaches a leaf
These are examples of a depth-first traversal.
Follow a path that descends the levels of a
tree as deeply as possible until reaches a leaf
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Traversals of a Tree
Level-order traversal: begin at the root, visit nodes one level at a time
The visitation order of a level-order traversal.
This is an example of a breadth-first
traversal.
Follow a path that explores an
entire level before moving to the next level.
This is an example of a breadth-first
traversal.
Follow a path that explores an
entire level before moving to the next level.
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Answer
In this binary tree, D = node, L = left, R = right
Preorder (DLR) traversal yields: A, H, G, I, F, E, B, C, D
Postorder (LRD) traversal yields: G, F, E, I, H, D, C, B, A
In-order (LDR) traversal yields: G, H, F, I, E, A, B, D, C
Level-order traversal yields: A, H, B, G, I, C, F, E, D
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Traversals of a General Tree
A general tree has traversals that are in Level order Preorder Postorder
Inorder traversal not well defined for a general tree
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Traversals of a General Tree
The visitation order of two traversals of a general tree: (a) preorder; (b) postorder.
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Java Interfaces for Trees
An interface that specifies operations common to all trees
public interface TreeInterface{ public Object getRootData();
public int getHeight();public int getNumberOfNodes();public boolean isEmpty();public void clear();
} // end TreeInterface
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Expression Trees for Algebraic Expression The root contains the binary operator and children contain the operands. Order of children matches the order of operands. Such binary tree is called
expression Trees. No parentheses since order of operations is captured by the shape of the expression tree
Expression trees for four algebraic expressions..
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Expression Trees Inorder traversal produces the original infix expression, but
without any parentheses Preorder traversal produces the prefix expression Postorder traversal produces the postfix expression (b) preorder: +*abc postorder: ab*c+
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Expression Trees
Algorithm for evaluating an expression tree in postorder traversal
Algorithm evaluate(expressionTree)if (root of tree is operand)
return operand
else{ firstOperand = evaluate(left subtree of expressionTree)
secondOperand = evaluate(right subtree of expressionTree)operator = the root of expressionTreereturn the result of the operation operator and its operands
firstOperand and secondOperand}
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Decision Trees Basis of Expert System
An expert system also known as a knowledge based system that contains some knowledge. It was first developed by researchers in artificial intelligence.
The most common form of expert systems is made up of a set of rules that analyze information.
Each parent in a decision tree is a question that has a finite number of response.
Each possible answer to the question corresponds to a child of the node.
Leave node are conclusions that have no children
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Decision Trees
Helps users solve problems, make decisions
A binary decision tree to solve TV problem.
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Decision Trees A possible Java interface for a binary decision
tree.
public interface DecisionTreeInterface extends BinaryTreeInterface{ /** Task: Gets the data in the current node.
* @return the data object in the current node */public Object getCurrentData();/** Task: Determines whether current node contains an answer.* @return true if the current node is a leaf */public boolean isAnswer();/** Task: Moves the current node to the left (right) child of the current node. */public void advanceToNo();public void advanceToYes();/** Task: Sets the current node to the root of the tree.*/public void reset();
} // end DecisionTreeInterface
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Binary Search Trees
We can traverse nodes in any tree, so searching a node is possible. The least efficient search will be sequential search.
However, a search tree organizes its data so that a search is more efficient
Binary search tree Nodes contain Comparable objects A node's data is greater than the data in the node's left
subtree A node's data is less than the data in the node's right
subtree
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Binary Search Trees
Two binary search trees containing the same names as the tree in previous slide
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Questions
How many different binary search trees can you form from the strings a, b, and c?
What are the heights of the shortest and tallest trees that you formed?
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Answer
How many different binary search trees can you form from the strings a, b, and c?
5 different structures What are the heights of the shortest and tallest trees
that you formed? 2 and 3
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Binary Search Trees
Search begins at the root, ends at either the target node or leaf node.
The number of comparison is the number of nodes along the path from the root to target node.
The height of a tree directly affects the length of the longest path from the root to a leaf and hence affects the efficiency of a worst-case search.
Searching a binary search tree of height h is O(h).
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Binary Search Trees
An algorithm for searching a binary search tree
Algorithm bstSearch(binarySearchTree, desiredObject)// Searches a binary search tree for a given object.// Returns true if the object is found.if (binarySearchTree is empty)
return falseelse if (desiredObject == object in the root of binarySearchTree)
return trueelse if (desiredObject < object in the root of binarySearchTree)
return bstSearch(left subtree of binarySearchTree, desiredObject)else
return bstSearch(right subtree of binarySearchTree, desiredObject)
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Heaps
A complete binary tree Nodes contain Comparable objects Each node contains no smaller (or no larger) than
objects in its descendants Maxheap
Object in a node is ≥ its descendant objects. Root node contains the largest data
Minheap Object in a node is ≤ descendant objects Root node contains the smallest data
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Question?
Can a binary search tree ever be a maxheap?
Can you think of any ADT we have learnt so far that can be implemented by maxheap?
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Parse Tree for Compiler These rules form a grammar for algebraic
expression, much like English language grammar. An algebraic expression is either a term or two terms
separated by + or – operator A term is either a factor or two factors separated by * or /
operator. A factor is either a variable or an algebraic expression
enclosed in parentheses. A variable is a single letter
Check the syntax of each statement by applying these rules, if we can the derivation can be given as a parse tree.
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Parse TreeA parse tree for the
algebraic expression
a * (b + c)1). An algebraic expression is either a term or two terms separated by + or – operator
2). A term is either a factor or two factors separated by * or / operator.
3). A factor is either a variable or an algebraic expression enclosed in parentheses.
4). A variable is a single letter
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Parse Tree
Parse tree should be a general tree so that I can accommodate any expression.
We are not restricted to algebraic expression. We can use parse tree to check the validity of
any string according to any grammar Compiler uses parse tree both to check the
syntax of a program and to produce executable code.