CSC 2300 Data Structures

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CSC 2300Data Structures Algorithms

• February 6, 2007
• Chapter 4. Trees

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Today Trees

• Tree basics
• Tree implementation
• Induction and trees
• Tree traversal
• Tree algorithm analysis
• Tree application file system

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Terminology
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Rooted Tree

• Recursive definition. A tree consists of a
  distinguished node r, called the root, and zero
  or more nonempty (sub)trees T1, T2, , Tk, each
  of whose roots are connected by a directed edge
  from r.
• Parent and child.
• The root r is the parent of each subtree root.

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Tree

• Definition of leaves?
• Which nodes are leaves?
• Definition of siblings?
• Which nodes are siblings?
• Grandparent and grandchild?

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Path and Length

• A path from node n1 to node nk is defined as a
  sequence of nodes n1, n2, , nk such that ni is
  the parent of ni1, for 1i
• The length of this path is the of edges on the
  path, namely k-1.
• Is there a path of length zero?
• There is exactly one path from the root to each
  node.

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Depth and Height

• The depth of node ni is the length of the unique
  path from the root to ni.
• What is the depth of the root?
• The height of ni is the length of the longest
  path from ni to a leaf.
• What is the height of a leaf?
• The height of a tree is equal to the height of
  its root.
• The depth of a tree is equal to the depth of the
  deepest leaf.
• How is the height of a tree related to its depth?

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Ancestor and Descendant

• If there is a path from ni to nk, then ni is an
  ancestor of nk and nk is a descendant of ni.
• If ni ? nk, then ni is a proper ancestor of nk
  and nk is a proper descendant of ni.

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Implementation of Trees

• Can you suggest an obvious way to implement a
  tree?
• Have in each node, besides its data, a link to
  each child of the node.
• Why is this approach possibly very inefficient?
• The of children per node may vary greatly and
  is not known in advance.
• Is there a better solution?

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Solution
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Induction and Trees
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Check Your Answer

• m 5, i ?, l ?

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Check Your Answer

• m 2, i ?, l ?

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Tree Traversals
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Time Analysis of Tree Algorithm
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Application

• One popular use of tree is the directory
  structure in operating systems, including UNIX
  and DOS.

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List Directory
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Time Analysis

• Line 1 is executed exactly once per node, since
  each name is output once.
• It follows that line 2 must also be executed once
  per node.
• Line 4 can be executed at most once for each
  child of each node. But the number of children is
  exactly one less than the number of nodes.
• The for loop iterates once per iteration of
  line 4, plus each time the loop ends.
• Thus, if there are N file names to be output,
  then the running time is O(N).

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Traversal Strategy

• What is the traversal strategy?
• In preorder traversal, work in a node is
  performed before (pre) its children are processed.

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Size of Each Directory

• Suppose we know the number of disk blocks taken
  by each file. How can we find the size of each
  directory?

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Postorder Traversal

• In a postorder traversal, the work at a node is
  performed after (post) its children are evaluated.