Lecture 17 Trees

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Lecture 17 Trees

• July 21st , 2003

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Tree--Definition

• A tree is an acyclic, connected graph with one
  node designated as the root of the tree.
• Special case
• nonrooted tree (free tree)
• Forest acyclic graph (not necessarily
  connected)
• Parent VS children.

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Definition (cont.)

• Depth
• Leaf
• Node
• Binary tree
• Full binary tree
• Complete binary tree (almost full).

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Tree -- Applications

• In data structure
• Binary search tree
• 2-3 tree
• In file management
• Files in computer are organized in a tree-like
  structure.
• In biology
• Family tree

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Binary tree representation

• Left child -right child representation
• Pointer representation.
• Example 24, practice 20

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Tree traversal algorithms

• Pre-order
• In-order
• Post-order
• Basic idea is recursion (any thing similar to
  induction?).

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A traversal algorithm

• ALGORITHM PREORDER
• Predorder (tree T)
• write (r)
• for i1 to t do
• Preorder(T)
• end for
• end Preorder

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Traversal--Examples

• Example 25.
• Practice 21
• How algebra expressions can be represented as
  binary trees?
• Infix notation
• Prefix notation (Polish notation).
• Postfix notation (reverse polish notation).
• Practice 22.

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Some properties of trees

• A tree with n nodes has n-1 arcs
• Any tree with n nodes, the total number of arc
  ends is 2n-2.
• A binary tree has at most 2d nodes at depth d.
• The number of leaves in any binary tree is 1 more
  than the number of nodes with two children.

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Exercise

• Exercises 5.2
• 3, 10,23, 28, 38, 40, 41