Chapter 8 General and Binary Trees

1
Chapter 8General and Binary Trees
2
What is tree?

• An important class of graphs is those that are
  structured as trees.
• A tree is an acyclic simple, connected graph.
• A tree contains no loops and no cycles there is
  no more than one edge between any pair of nodes.

3
General Trees

• A tree is a finite set of zero or more nodes
  (v1,v2,,vn) such that
• There is one specially designated node (say v1)
  called Root(T).
• The remaining nodes (v2,,vn) are partitioned
  into disjoint sets named T1,T2,,Tm such that
  each Ti is itself a tree.

Tree with directed edges
4
General Trees

• Root(T) A
• The root has no incoming branches or in-degree
  0
• Three subtrees are rooted at B, C and D.
• Leaves of the tree are E, F, K, L, H, I, and J.
• The leaves have out-degree 0

5
General Trees

• A is the parent of B, and C.
• B is a child of A.
• C is a child of A.
• B and C have the same parent, so B and C are
  called brothers (or sisters)

• The nodes of a tree are said to be on level,
  where a nodes level is determined by the length
  of the path from the root to that node.
• Level 0 A
• Level 1 B, C
• Level 2 E,F,G
• The height of a tree is one plus the number of
  the highest level on which there are node.
• Height 3
• The weight of a tree is its number of leaf nodes.
  
• Weight 3

6
General Trees

• Level ?
• Height ?
• Weight ?
• Red color subtree
• Parent ?
• Child ?
• Brothers?
• Leaves?

A
D
C
B
H
F
G
I
J
E
K
L
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General Trees

• Collection of rooted tree is called a forest.

8
Forms of Representation

• We have used one graphical notation to represent
  trees.
• Other graphical notations for tree structure
  include
• Nested sets
• Nested parentheses
• Indentation

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Forms of Representation

• Nested Set

B
D
E
H
J
A
F
I
K
C
G
L
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Forms of Representation

• Nested Parentheses
• (A(B(E))(C(F)(G(K)(L)))(D(H)(I)(J)))

• Indentation
• A-------------------
• B-------------
• E----------
• C--------------
• F-----------
• G----------
• K-----
• L-----
• D-------------
• H---------
• I----------
• J---------

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Forms of Representation

• Nested set ?
• Nested parentheses ?
• Indentation ?

12
Binary Trees

• A binary tree is a finite set of nodes which
  either is empty or contains two disjoint binary
  trees that are called its left and right
  subtrees.
• The maximum out-degree of any node of a binary
  tree is 2. One is left node and another one is
  right node.
• Tree a and tree b are not the same. Tree a with
  left subtree and tree b with right subtree.
• Tree c is not a binary tree, because its subtree
  does not have leftness or rightness

(c)
(a)
(b)
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Binary Trees

• Two binary trees are said to be similar if they
  have the same structure.
• Two binary trees are said to be equivalent if
  they are similar and contain the same
  information.

• Tree 1 and Tree 2 are _____
• Tree 1 and Tree 3 are _____

Tree 3
Tree 2
Tree 1
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Binary Trees

• A binary tree is said to be complete if it
  contains the maximum number of nodes possible for
  its height.
• A complete binary tree with K levels will have 2K
  1 nodes.
• For example, the following tree with 4 levels
  will have 16-1 15 nodes

Level 0
Level 1
Level 2
Level 3
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Binary Trees

• A binary tree with K level is said to be almost
  complete if level 0 through K-2 are full and
  level K-1 is being filled left to right.

(K-2)
(K-1)
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Binary Trees

• Is this a complete binary tree?
• Is this an almost complete binary tree?

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Representation of Binary Trees

• Binary trees are most commonly represented by
  linked list.
• Each node in linked list can be regarded as
  having three elementary fields
• An information area
• Pointers to the left
• Pointers to the right

The topological placement of the boxes of the
linked lists on the page is of course
meaningless Storage for the nodes could be
scattered throughout memory.
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Binary Tree Representation of General Trees

• It is considerably easier to represent binary
  trees in programs than it is to represent general
  trees.
• With a general tree, it is unpredictable how many
  edges will emanate from a node at any given time.
• Binary trees are attractive in that each node has
  a predictable maximum number of subtree pointers
  2.

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Binary Tree Representation of General Trees

• The algorithm for converting a general tree to a
  binary tree form
• Inserting edges connecting siblings and delete
  all of a parents edges to its children except to
  its leftmost offspring.

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Binary Tree Representation of General Trees

• Rotate the resultant diagram 45 degree to
  distinguish between left and right subtrees.

Left pointers are always from a parent node to
its first (leftmost) child in the original
general tree. Right pointers are always from a
node to one of its siblings in the original tree.
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Binary Tree Representation of General Trees

• Convert the following general tree to binary tree.

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Binary Tree Representation of General Trees

• A forest of general tree can be converted to a
  single binary tree by considering the roots to be
  siblings.

A
H
C
A
B
C
D
I
F
E
D
H
B
E
I
K
G
J
G
F
J
K
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Binary Tree Representation of General Trees

• Construct a binary tree from the following forest.

I
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Example Trees

• Trees are commonly used to structure data to
  facilitate searches for particular nodes.
• Trees are also useful for representing
  collections of data that have branching logical
  structures. For example, trees representing
  arithmetic statements and a collection of data.

Tree 1 cde
Tree 3
Tree 2 (((ab)c/d)e?f)/g

• Searching for a node in the tree can be
  considerably faster than searching the same
  collection sequentially.

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Binary Search Trees

• An important application of binary trees is their
  use in searching.
• The property that makes a binary tree into a
  binary search tree is that for every node, X, in
  the tree, the value of all the items in its left
  subtree are smaller than the item in X, and the
  values of the items in its right subtree are
  larger than the item in X.

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Binary Search Trees

• Are they binary search trees?

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Sequential Searches

• The process of going through a tree in such a way
  that each node is visited once and only once is
  called tree traversal.
• When a tree is traversed, its entire collection
  of nodes is looked at. There are several
  well-known methods of binary tree traversal. Each
  imposes a sequential, linear ordering upon the
  nodes of a tree.
• A node is said to be visited when it is
  encountered in the traversal whatever processing
  is desirable on its contents is done at that
  time.
• There are three kinds of basic activities in each
  of the binary tree traversal algorithms
• Visit the root
• Traverse the left subtree
• Traverse the right subtree
• The order in which these type of activities are
  performed leads to three traversal method.

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Sequential Searches

• Pre-order traversal
• Visit the root.
• Traverse the left subtree
• Traverse the right subtree
• In-order traversal
• Traverse the left subtree
• Visit the root.
• Traverse the right subtree
• Post-order traversal
• Traverse the left subtree
• Traverse the right subtree
• Visit the root.

1
cde
2
4
3
4
3
cde
2
1
4
3
cde
2
1
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Sequential Searches

• Pre-order traversal results in a string
  representation in prefix form an operator
  precedes its operands.
• In-order traversal results in a string
  representation in infix form an operator appears
  between its operands.
• Post-order traversal results in a string
  representation in postfix form an operator is
  preceded by its two operands.

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Sequential Searches

• Find the results of pre-order traversal, in-order
  traversal, and post-order traversal of the
  following two trees.

• Pre-order ?
• In-order ?
• Post-order ?

• Pre-order ?
• In-order ?
• Post-order ?

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Direct Searches

• The properties of a binary search tree imply that
  there is a procedure for determining whether or
  not a node with a given key resides in the tree
  and for finding that node when it exists.
• To find the node with key k in the binary search
  tree rooted at Ri, the following steps are taken.
• If the tree is empty, the search terminates
  unsuccessfully.
• If k Ki, the search terminates successfully
  the sought node is Ri.
• If k lt Ki, the left subtree of Ri is searched.
• If k gt Ki, the right subtree of Ri is searched.
• Finding a particular node in a tree that is not a
  binary search tree is usually done with a
  sequential search rather than a direct search,
  unless the tree is specially structured and the
  nodes have key fields.
• The effort required to find a particular record
  in a binary search tree depends upon the position
  of the record in the tree. The farther the sought
  record is from the trees root, the greater is
  the effort required to find the record.

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Direct Searches

• The search effort to find particular node in a
  binary search tree is measured by the number of
  comparisons made before the search terminates,
  either successfully or unsuccessfully.

(d)
(c)
(b)
(a)

• For example, to find the record with name DOG in
  the above binary search trees require
• Tree (a) 1 comparison
• Tree (b) 4 comparisons
• Tree (c) 3 comparisons
• Tree (d) 2 comparisons
• If it were necessary often to find DOG, then tree
  (a) would probably be considered to be a better
  binary search tree than would tree (b).

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Direct Searches

• A binary search tree cannot be evaluated well on
  the sole basis of the search path that it
  provides to just one record. Rather, the search
  paths through the entire tree should enter into
  the evaluation thus the expected (i.e., weighted
  average) length of a search path in the tree is
  useful.
• If the access probabilities for the keys
  structured in the binary search trees were
• APE 0.2
• BEE 0.4
• COW 0.3
• DOG 0.1
• The expected search lengths for binary search
  tree (a), (b), (c), and (d) would be
• APE BEE COW DOG
• (a) 0.24 0.43 0.32 0.11 2.7
• (b) 0.21 0.42 0.33 0.14 2.3
• (c) 0.22 0.41 0.32 0.13 1.7
• (d) 0.22 0.41 0.33 0.12 1.9
• Note that here in order to reduce the expected
  search length, the tree is structured so that the
  most frequently accessed names are placed as
  close as possible to the root.

comparison
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Balancing Binary Search Trees

• If the access probabilities were known and no
  insertions were to be made in the tree, the
  binary search tree that yields optimal
  performance could be constructed.
• However, if the access probabilities are
  unavailable, balancing tree give nearly optimal
  expected search lengths.

• Inserting ZEBRA would be easy it would be the
  right subtree of GIRAFFE and the tree would still
  be in balance.
• Instead, however, try to insert APE. The position
  of APE will be as the left subtree of BEE, which
  pulls the tree out of balance. To make tree
  balance, all of the nodes of the original tree
  had to be moved.

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Height-Balanced (AVL) Trees

• One type of almost completely balanced trees is
  known as the height-balanced tree, or the AVL
  tree.
• A tree is height-balanced if the height of the
  left subtree of Ri, and the height of right
  subtree of Ri differ by at most 1.

(c)
(b)
(a)
Height-balanced trees

• All the above trees are height-balanced. Note
  that although it is height-balanced, tree (c) is
  not completely balanced.

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Height-Balanced (AVL) Trees
(a)
(b)
(c)
Binary trees that are not height-balanced

• The height constraint is violated at node APE in
  tree (a), at node COW in tree (b), and at node
  FOX in tree (c).
• AVL trees provide good direct search performance.
  Although they tend to look somewhat more sparse
  than do completely balanced trees, the search
  they require are only moderately longer.