Lecture 12: Graph Theory Tree

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Lecture 12 Graph Theory - Tree

• Introduction to Trees
• Basic Definitions
• Rooted Trees
• Applications of Trees
• Trees as Models
• Binary Search Trees
• Tree Traversal

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12.1. Basic Definitions

• A tree
• Is a connected graph with no cycles
• It can not contain multiple edges or loops.
• Theorems
• A graph is a tree if and only if there is a
  unique simple path between any two of its
  vertices. (Why?)
• VE1 (Why?)

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12.1. Basic Definitions

• Examples
• G1 and G2 are trees.
• G3 and G4 are not a tree.

G4
G1
G2
G3
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12.1. Basic Definitions

• A forest
• Is a graph where each of its connected components
  is a tree.

A graph with three connected components
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12.1. Basic Definition

• In chemistry
• Representing molecules
• Each vertex represents an atom.
• Each edge represents the bond between the atoms.
• Examples

Isobutane
Butane
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12.2. Rooted Trees

• A rooted tree
• Is a directed graph T satisfying two conditions
• It is a tree if the directions of the edges are
  ignored.
• There is a unique vertex r such that the indegree
  of r is 0 and the indegree of any other vertex is
  1.
• This vertex r is called the root of the rooted
  tree.
• We can change any unrooted tree into a rooted
  tree by choosing any vertex as the root.

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12.2. Rooted Trees

• Examples
• Different choices of the root produce different
  rooted trees.

With root a
T
With root c
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12.2. Rooted Trees

• Examples
• A family tree is a rooted tree where
• the vertices represents family members and
• the edges represent parent-child relationship.

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12.2. Rooted Trees

• In business
• Representing the structure of a large
  organization
• Each vertex represents a position.
• Each edge indicates that the person represented
  by the initial vertex is the (direct) boss of the
  person represented by the terminal vertex.
• Examples

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12.2. Rooted Trees

• In computer science
• Representing computer file systems
• The root represents the root directory.
• Each internal vertex represents a subdirectory.
• A leave represents an ordinary file or empty
  subdir.
• Examples

The root is the root directory / Internal
vertices are directories Leaves are files.
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12.2. Rooted Trees

• Rooted tree terminology
• For a directed edge from vertex u to vertex v
• u is a parent of v.
• v is a child of u.
• For a vertex u
• Any vertices (excluding vertex u) on the path
  from the root to the vertex u are the ancestors
  of u.
• Vertex u is the descendant of these vertices.
• A terminal vertex or a leaf is a vertex that has
  no children. (There must be at least one leaf.
  Why?)
• An internal vertex is one that has children.
• A subtree is a subgraph of a tree.

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12.2. Rooted Trees

• Examples
• Find the followings
• Children of b?
• Parent of b?
• Ancestors of f?
• Descendant of a?
• Terminal vertices (leaves)?
• Internal vertices?

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12.2. Rooted Trees

• Examples
• A tree T and its subtree rooted at g.

T
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12.2. Rooted Trees

• Rooted tree terminology (continued)
• An m-ary tree
• Is a rooted tree in which each internal vertex
  has at most m children.
• Special case binary tree where m 2.
• A complete m-ary tree
• Is a rooted tree in which each internal vertex
  has exactly m children.
• A balanced m-ary tree with height h
• Is a m-ary tree with all leaves being at levels h
  or h-1.
• Level distance from root
• Height maximum level

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12.2. Rooted Trees

• Examples
• T1 is a complete binary tree. T1 is not balanced.
• T2 is a balanced complete 5-ary tree.
• T3 is a balanced 3-ary but not a complete m-ary
  tree for any m.

T3
T2
T1
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12.3. Ordered Rooted Trees

• Whats the total number of nodes (vertices) for a
  complete balanced binary tree of height h?
• ?ltNlt?

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12.3. Ordered Rooted Trees

• An ordered rooted tree
• Is a rooted tree where the children of each
  internal vertex are ordered.
• In an ordered binary tree, if an internal vertex
  u has two children
• The first child is called the left child.
• The tree rooted at the left child is called the
  left subtree of vertex u.
• The second child is called the right child.
• The tree rooted at the right child is called the
  right subtree of vertex u.

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12.2. Rooted Trees

• Examples
• The left child of d is f.
• The right child of d is g.
• (b) and (c) show the left and right subtrees of c.

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12.5. Tree Traversal

• Tree traversal
• Is a procedure that systematically visits every
  vertex of an ordered rooted tree.
• Visiting a vertex processing the data at the
  vertex.
• Three most commonly used algorithms
• Preorder traversal.
• Inorder traversal.
• Postorder traversal.

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12.5.1. Preorder Traversal

• Applications print a structured document

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12.5.2. Inorder Traversal

• Applications print out information stored in a
  binary search tree.

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2
8
1
7
9
4
3
5
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12.5.3. Postorder Traversal

• Applications compute space used by files in a
  directory and its subdirectories.

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cs16/
8
3
7
todo.txt1K
homeworks/
programs/
4
5
6
1
2
DDR.java10K
Stocks.java25K
h1c.doc3K
h1nc.doc2K
Robot.java20K
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12.5. Tree Traversal

• Let T be an ordered rooted tree with root r.
• Suppose that T1, T2, ?, Tn are the subtrees at r
  from left to right in T.

T2
T3
T1
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12.5.1. Preorder Traversal

• Preorder traversal
• Begins by visiting r.
• Next traverses T1 in preorder, then T2 in
  preorder, ?, then Tn in preorder.
• Ends after Tn has been traversed.

Preorder traversal
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12.5.1. Preorder Traversal

• Preorder traversal
• An example of recursive algorithm.

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12.5.1. Preorder Traversal

• Examples
• Preorder traversal
• Visit root, visit subtrees left to right.

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12.5.1. Preorder Traversal
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12.5.1. Preorder Traversal

• In binary tree the traversal is characterized by
• Visiting a parent before its children and a left
  child before a right child.
• Algorithm
• Step 1 Visit the root.
• Step 2 Go to the left subtree.
• If one exists, do a preorder
  traversal.
• Step 3 Go to the right subtree.
• If one exists, do a preorder
  traversal.

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12.5.1. Preorder Traversal

• Applications print a structured document

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12.5.2. Inorder Traversal

• Inorder traversal
• Begins by traversing T1 in inorder.
• Next visits r, then traverses T2 in inorder, ?,
  then Tn in inorder.
• Ends after Tn has been traversed.

In order traversal
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12.5.2. Inorder Traversal

• Examples
• Inorder traversal
• Visit leftmost subtree, visit root, visit other
  subtrees left to right.

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12.5.2. Inorder Traversal
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12.5.2. Inorder Traversal

• Applications print out information stored in a
  binary search tree.

6
2
8
1
7
9
4
3
5
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12.5.3. Postorder Traversal

• Postorder traversal
• Begins by traversing T1 in postorder.
• Next traverses T2 in postorder, then traverses T3
  in postorder, ?, then Tn in postorder.
• Ends by visiting r.

Postorder traversal
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12.5.3. Postorder Traversal

• Examples
• Postorder traversal
• Visit subtrees left to right, visit root.

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12.5.3. Postorder Traversal
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12.5.3. Postorder Traversal

• Applications compute space used by files in a
  directory and its subdirectories.

9
cs16/
8
3
7
todo.txt1K
homeworks/
programs/
4
5
6
1
2
DDR.java10K
Stocks.java25K
h1c.doc3K
h1nc.doc2K
Robot.java20K
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12.5. Tree Traversal

• Shortcut to list the vertices in preorder,
  inorder, postorder
• Draw a curve around the ordered rooted tree
  starting at the root and move along the edges.

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12.5. Tree Traversal

• Preorder list is obtained by
• listing each vertex the first time this curve
  passes it.
• Result a, b, d, h, e, i, j, c, f, g, k.
• Inorder list is obtained by
• listing a leaf the first time the curve passes it
  and
• listing each internal vertex the second time the
  curve passes it.
• Result h, d, b, i, e, j, a, f, c, k, g.
• Postorder list is obtained by
• listing a vertex the last time it is passes on
  the way back up to its parent.
• Result h, d, i, j, e, b, f, k, g, c, a.

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12.4. Binary Search Trees

• Ordered rooted trees are often used to store
  information.
• A binary search tree
• Is a binary tree in which each vertex is labeled
  with a key such that
• No two vertices have the same key.
• If vertex u belongs to the left subtree of vertex
  v, then u ? v.
• If vertex w belongs to the right subtree of
  vertex v, then v ? w.

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12.4. Binary Search Trees

• Binary search tree construction algorithm
• Start with a tree containing just one vertex (the
  root).
• Let v the root.
• To add a new item a, do the following until stop
• If a lt v
• If v has a left child then v left child of v
  (move to the left)
• Else add a new left child to v with this item as
  its key. Stop.
• If a gt v
• If v has a right child then v right child of v
  (move to the right).
• Else add a new right child to v with this item as
  its key. Stop.

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12.4. Binary Search Trees

• Examples
• Construct a binary search tree for the list 5, 9,
  8, 1, 2, 4, 10, 6.

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12.4. Binary Search Trees

• Binary search tree search algorithm
• Let v the root.
• To search the item a, do the following until
  stop
• If a v then stop.
• If a lt v
• If v has a left child then v left child of v
  (move to the left)
• Else stop.
• If a gt v
• If v has a right child then v right child of v
  (move to the right).
• Else stop.

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12.4. Binary Search Trees

• Examples
• Search for 7 in the following binary tree.
• If not found, add it to the tree.

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12.5. Tree Traversal

• How many comparisons best case, worst case?

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12.6. Further Readings

• Introduction to Trees
• Basic Definitions Section 12.1.
• Rooted Trees Section 12.2.
• Applications of Trees
• Tree Traversal Section 12.2.