CS210- Lecture 11
June 23, 2005
Agenda
 Applications of Traversals
 Expression Tree Evaluation
 Data Structures for representing trees
 A Vector based structure for Binary Trees
 A Linked structure for Binary Trees
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Expression Tree Evaluation
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A postorder traversal of the binary tree can
be used to solve the expression tree
evaluation problem.
In this problem, we are given an arithmetic
expression tree, that is, a binary tree where
each external node has a value associated
with it and each internal node has an
arithmetic operation associated with it and we
want to compute the value of the arithmetic
expression represented by the tree.
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Picture of a Expression tree
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Algorithm to evaluate expression tree
Algorithm evaluateExpression(T, v):
if T.isInternal(v) then
let o be the operator stored at v
X <- evaluateExpression(T, T.left(v))
Y <- evaluateExpression(T, T.right(v))
return x o y
else
return the value stored at v.
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Application of inorder traversal
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Most important application of inorder
traversal arises when we store an
ordered sequence of elements in a
binary tree, defining a structure called
binary search tree.
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Binary Search Trees
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Let S be a set whose elements have an order
relation. For example, S could be a set of
integers. A binary search tree for S is a
proper binary tree such that:
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Each internal node v of T, stores an element of S,
denoted by x(v).
For each internal node v of T, the elements stored
in the left subtree of v are less than or equal to
x(v) and the elements stored in the right subtree
of v are greater than or equal to x(v).
The external nodes of T do not store any element.
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Binary Search Tree
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An inorder traversal of the internal nodes of a binary
search tree T visits the elements in nondecreasing
order.
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A Vector based structure for binary trees
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A Simple structure for representing a
binary tree T is based on a way of
numbering the nodes of T. For every
node v of T, let p(v) be the integer
defined as follows:
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If v is the root of T, then p(v) = 1
If v is the left child of node u, then p(v) =
2p(u).
If v is the right child of node u, then p(v) =
2p(u) + 1.
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A Vector based structure for Binary Trees
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What if parent index
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Parent ?
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A Vector based structure for Binary Trees
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A linked structure for Binary Trees
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A natural way to realize a binary tree T
is to use a linked structure.
parent
right
left
element
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A linked structure for Binary trees
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root
size
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