Trees

1

• Trees

2
Trees

• Definition A tree is a connected undirected
  graph with no simple circuits.
• Since a tree cannot have a simple circuit, a tree
  cannot contain multiple edges or loops.
• Therefore, any tree must be a simple graph.
• Theorem An undirected graph is a tree if and
  only if there is a unique simple path between any
  of its vertices.

3
Trees

• Example Are the following graphs trees?

Yes.
No.
No.
Yes.
4
Trees

• Definition An undirected graph that does not
  contain simple circuits and is not necessarily
  connected is called a forest.
• In general, we use trees to represent
  hierarchical structures.
• We often designate a particular vertex of a tree
  as the root. Since there is a unique path from
  the root to each vertex of the graph, we direct
  each edge away from the root.
• Thus, a tree together with its root produces a
  directed graph called a rooted tree.

5
Applications of Trees

• There are numerous important applications of
  trees, only three of which we will discuss today
• Network optimization with minimum spanning
  trees
• Problem solving with backtracking in decision
  trees
• Data compression with prefix codes in Huffman
  coding trees

6
Tree Terminology

• If v is a vertex in a rooted tree other than the
  root, the parent of v is the unique vertex u such
  that there is a directed edge from u to v.
• When u is the parent of v, v is called the child
  of u.
• Vertices with the same parent are called
  siblings.
• The ancestors of a vertex other than the root are
  the vertices in the path from the root to this
  vertex, excluding the vertex itself and including
  the root.

7
Tree Terminology

• The descendants of a vertex v are those vertices
  that have v as an ancestor.
• A vertex of a tree is called a leaf if it has no
  children.
• Vertices that have children are called internal
  vertices.
• If a is a vertex in a tree, then the subtree with
  a as its root is the subgraph of the tree
  consisting of a and its descendants and all edges
  incident to these descendants.

8
Tree Terminology

• The level of a vertex v in a rooted tree is the
  length of the unique path from the root to this
  vertex.
• The level of the root is defined to be zero.
• The height of a rooted tree is the maximum of the
  levels of vertices.

9
Terminology

• The degree of a node is the number of subtrees of
  the node
• The node with degree 0 is a leaf or terminal
  node.
• A node that has subtrees is the parent of the
  roots of the subtrees.
• The roots of these subtrees are the children of
  the node.
• Children of the same parent are siblings.
• The ancestors of a node are all the nodes along
  the path from the root to the node.

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Tree Properties
Property Value Number of nodes Height Root
Node Leaves Interior nodes Number of
levels Ancestors of H Descendants of B Siblings
of E Right subtree
A
B
C
D
E
F
G
I
H
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Trees

• Example I Family tree

James
Christine
Bob
Frank
Joyce
Petra
12
Trees

• Example II File system

/
usr
temp
bin
bin
spool
ls
13
Trees

• Example III Arithmetic expressions

?

-
y
z
x
y
This tree represents the expression (y z)?(x -
y).
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Trees

• Definition A rooted tree is called an k-ary tree
  if every internal vertex has no more than k
  children.
• The tree is called a full k-ary tree if every
  internal vertex has exactly k children.
• An k-ary tree with k 2 is called a binary tree.
• Theorem A tree with n vertices has (n 1)
  edges.
• Theorem A full k-ary tree with i internal
  vertices contains n ki 1 vertices.

15
Binary Search Trees

• If we want to perform a large number of searches
  in a particular list of items, it can be
  worthwhile to arrange these items in a binary
  search tree to facilitate the subsequent
  searches.
• A binary search tree is a binary tree in which
  each child of a vertex is designated as a right
  or left child, and each vertex is labeled with a
  key, which is one of the items.
• When we construct the tree, vertices are assigned
  keys so that the key of a vertex is both larger
  than the keys of all vertices in its left subtree
  and smaller than the keys of all vertices in its
  right subtree.

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Binary Search Trees

• Example Construct a binary search tree for the
  strings math, computer, power, north, zoo,
  dentist, book.

math
17
Binary Search Trees

• Example Construct a binary search tree for the
  strings math, computer, power, north, zoo,
  dentist, book.

math
computer
18
Binary Search Trees

• Example Construct a binary search tree for the
  strings math, computer, power, north, zoo,
  dentist, book.

math
power
computer
19
Binary Search Trees

• Example Construct a binary search tree for the
  strings math, computer, power, north, zoo,
  dentist, book.

math
power
computer
north
20
Binary Search Trees

• Example Construct a binary search tree for the
  strings math, computer, power, north, zoo,
  dentist, book.

math
power
computer
north
zoo
21
Binary Search Trees

• Example Construct a binary search tree for the
  strings math, computer, power, north, zoo,
  dentist, book.

math
power
computer
north
zoo
dentist
22
Binary Search Trees

• Example Construct a binary search tree for the
  strings math, computer, power, north, zoo,
  dentist, book.

math
power
computer
north
zoo
dentist
book
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Binary Search Trees

• To perform a search in such a tree for an item x,
  we can start at the root and compare its key to
  x. If x is less than the key, we proceed to the
  left child of the current vertex, and if x is
  greater than the key, we proceed to the right
  one.
• This procedure is repeated until we either found
  the item we were looking for, or we cannot
  proceed any further.
• In a balanced tree representing a list of n
  items, search can be performed with a maximum of
  ?log(n 1)? steps (compare with binary search).

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Representation of Tree Nodes

• Every tree node has the following
• Object or data useful information
• Children pointers to its children nodes

O
O
O
O
O
25
Left Child - Right Sibling
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Tree Implementation

• typedef struct tnode
• int key
• struct tnode lchild
• struct tnode sibling
• ptnode
• Create a tree with three nodes (one root two
  children)
• Insert a new node (in tree with root R, as a new
  child at level L)
• Delete a node (in tree with root R, the first
  child at level L)

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Binary Trees

• A special class of trees max degree for each
  node is 2
• Recursive definition A binary tree is a finite
  set of nodes that is either empty or consists of
  a root and two disjoint binary trees called the
  left subtree and the right subtree.
• Any tree can be transformed into binary tree.
• by left child-right sibling representation

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Example
A
B
C
E
D
G
F
K
H
L
I
M
J
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Node Structure for Binary Trees
Left data Right
typedef struct TreeNode PtrToNode typedef
struct PtrToNode Tree struct TreeNode
ElementType data Tree
left Tree right

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Tree Traversal

• Three main methods
• Preorder
• Postorder
• Inorder
• Recursive definition
• PREorder
• Visit the root
• Traverse in preorder the children (subtrees)
• POSTorder
• Traverse in postorder the children (subtrees)
• Visit the root
• INorder
• Traverse inorder the left child (subtree)
• Visit the root
• Traverse inorder the rest of the subtrees

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Preorder

• preorder traversal
• Algorithm preOrder(v)
• visit node v
• for each child w of v do
• recursively perform preOrder(w)

A
B
C
D
E
F
G
A B D E G H I F C
I
H
32
Preorder Traversal (recursive version)
void preorder(ptnode ptr) / preorder tree
traversal / if (ptr)
printf(d, ptr-data)
preorder(ptr-left) predorder(ptr-right)

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Postorder

• postorder traversal
• Algorithm postOrder(v)
• for each child w of v do
• recursively perform postOrder(w)
• visit node v

A
B
C
D
E
F
G
D H I G E F B C A
I
H
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Postorder Traversal (recursive version)
void postorder(ptnode ptr) / postorder tree
traversal / if (ptr)
postorder(ptr-left) postdorder(ptr-righ
t) printf(d, ptr-data)
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Inorder

• inorder traversal
• Algorithm inOrder(v)
• for left child w of v do
• recursively perform inOrder(w)
• visit node v
• for remaining child vertices u do
• recursively perform inOrder(u)

A
B
C
D
E
F
G
D B H G I E F A C
I
H
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Inorder Traversal (recursive version)
void inorder(ptnode ptr) / inorder tree
traversal / if (ptr)
inorder(ptr-left) printf(d,
ptr-data) indorder(ptr-right)
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Program 7

• Implement a binary search tree. Build it using
  the representation on page 97. Your binary tree
  will hold 30-character strings. It will
  initially be empty, and the user will be prompted
  for data to insert in the tree. The following
  options will be provided in your program
• insert data include a file option
• preorder printout
• inorder printout
• postorder printout