knure, software department, ph. 7021-446, e-mail: [email protected]

KNURE, Software department,Ph. 7021-446, e-mail: [email protected]

NN..VV. . BilousBilous

Faculty of computer sciencesSoftware department, KNURE

The trees.The trees.

Discrete mathematicsDiscrete mathematics..

N.V.Belous 7.Introduction to Trees 2

The basic definitionsThe basic definitions

A tree is a connected undirected graph with no simple circuits.

Graphs containing no simple circuits that are not necessarily connected are called forests .

d

ba

c

fg G1G2 G3 G4

The graph G1 is a Tree , Subgraphs are form the forest.42 GG



A particular vertex of a tree is designated as the root.

A tree together with its root produces a directed graph called a rooted tree.

h

e

i

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b

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c

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a

A Rooted Tree: a – a root



Let’s T is a rooted tree. If v is a vertex in T 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 a child of u.

Example

The parent of h and i is g.

The children of g are h, and i i

g

k

h



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.

Example

The sibling of h is i.

The ancestors of e are c, b, and a.

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b

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c

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a



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 (exclude a root) are called

internal vertices. Example

The descendants of b are c, d and e.

The leaves are d, e, f, i and k. The internal vertices are a, b, c, g and h.

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i

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k

b

d

c

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a



If a is a vertex in a tree, 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.

A Rooted Tree and its Subtree.

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A spainning tree is a tree which contains all vertices of the initial graph.

The chord of spainning tree Н in graph G is the edge of graph G, which are not belonging to spainning tree Н (this edge, is between vertices 3 and 4).

Spainning tree H of graph G.


Cayley’s ProblemCayley’s Problem

Let М is set of cities.с(a, b) is cost of road construction between

cities a and b (a, b M). What should be net of roads between the cities a

and b (a, b M) that on it was possible to pass from any city a into any city b and that cost of this net с(a, b) was minimal?


Cayley’s ProblemCayley’s Problem

It is possible to construct nn-2 trees on n vertices.

n=4 , then 42=16.

Quantity of trees with 4 vertices.


Tree of the minimal weight (cost)Tree of the minimal weight (cost)

Graph that have a number assigned to each edge is called a weighted graphs.

A number assigned to each edge of graph is called

a weight.

The sum of edges weights for a graph is called a weight of graph from G .

Graph G.


Algorithm for finding the minimal weight treeAlgorithm for finding the minimal weight tree

1. Arrange edges use the order of their weight number.

2. Include in the spainning tree edges use the increasing order of weight number.

3. Add a next edge. If an added edge generate the circuit in the graph then delete it from the tree.

4. The tree is constructed when in it is included n-1 edge.



Example.

It is given graph G. To construct for it a tree of the minimal weight.



Continuation of example.СЕ = 4 АВ = 6

АС = 7 SН=6+8+7+4=25.АD = 8


Algorithm of tree codingAlgorithm of tree coding

1)Count the number of vertices n in the tree. The length of the tree code is equal to n-2.

2) Write all leaves of the tree. From of all leaves find the leaf with a minimal number. This leaf is deleted from the tree with the incidence edge. The number of an adjacence vertex write in the tree code.

3) This process repeats until will receive the code by length n-2.


Example of coding of treesExample of coding of trees

2 4

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78

9 11

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2 4

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9 11

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2 4

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9 11

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2 4

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78

9 11

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Algorithm of the tree decodingAlgorithm of the tree decoding

1) Consider the tree code. If the code length is equal top, then the number of vertices in the tree is equal to p+2;

2) Find vertices (leaves number that do not enter into a code);

3) Find the leaf with minimal number and connect it with the first unuseable vertex in the code. If the connected vertex does not use in the next positions of the code, write this vertex in the sequence of leaves;

4) Repeat point 3),until use all leaves;

5) The last vertex in the code connect with the leaf with maximal number.


Algorithm of decoding of treesAlgorithm of decoding of trees

Example.The code: 1, 1, 1, 5. It is necessary to construct a tree.The decision.

1. Total of vertices equally 4+2=6.2. Find leaves: 2, 3, 4, 6

2 3 4

1 5

6


The level and height of treeThe level and height of tree

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.


The level and height of treeThe level and height of tree

Example.

The root a is at level 0.

b, j, o and k are at level 1.

d, g, i, m, n, and s are at level 3.

c, e, f, l and p are at level 2.

h is at level 4.

This tree has height 4.

a

k

l p

m

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kd g

c

b

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j

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in


m-ary tree, binary treem-ary tree, binary tree

A rooted tree is called an m-ary tree if every internal vertex has no more than m children.

The tree is called a full m-ary tree if every internal vertex has exactly m children.

An m-ary tree with m = 2 is called a binary tree.

A Binary Tree

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j lk


A binary tree.A binary tree.

In an ordered binary tree, if an internal vertex has two children, the first child is called the left child and the second child is called the right child.

The tree rooted at the left child of a vertex is called the left subtree of this vertex.

The tree rooted at the right child of a vertex is called the right subtree of the vertex .

N.V.Belous 8.Properties of Trees 23

Properties of TreesProperties of Trees

Properties of Trees

1. An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices.

2. A tree with n vertices has n –1 edges.

3. A full m-ary tree with i internal vertices contains n = mi + 1 vertices


Properties of TreesProperties of Trees

4. A full m-ary tree

(I) with n vertices has i = (n - 1)/m internal vertices and

l = [(m - 1)n + 1]/m leaves.

(II) with i internal vertices has n = mi + 1 vertices and

l = (m - 1)i + 1 leaves.

(III) with l leaves has n = (ml - 1)/(m - 1) vertices and i = (l-1)/(m-1) internal vertices.


Infix, prefix, and postfix notationInfix, prefix, and postfix notation

An ordered rooted tree can be used to represent such expressions, where the internal vertices represent operations, and the leaves represent the variables or numbers.

There are infix, prefix, and postfix notation ordered rooted tree.



Infix notation: any operation write between operands.

Prefix notation: at first write operations, at second – operands.

Postfix notation: at first write operands, at second – operations.



The prefix form:

+ + x y 2 / – x 4 3

The infix form:

x+y2+x–4/3x+y/2

The postfix form:

x y + 2 x 4 – 3 / + y

+

x

-

x 4

/

32

*

+

The Binary Tree for ((x+y)2)+((x– 4)/3)


Tree TraversalTree Traversal

T1,T2,…Tn are the subtrees at r from left to right

in T. The preorder traversal begins by visiting r. It continues by traversing T1 in preorder, then T2 in

preorder, and so on, until Tn is traversed in preorder. Step 1: Visit r

T1 T2 Tn

Step 2: Visit T1

in preorder

Step 3: Visit T2

in preorder

Step n+1: Visit Tn

in preorder

r


The preorder traversalThe preorder traversal

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Algorithm of Preorder TraversalAlgorithm of Preorder Traversal

Algorithm Preorder Traversal.procedure preorder (T ordered rooted tree)r = root of Tlist rfor each child c of r from left to rightbegin T(c) = subtree with c as its root preorder(T(c))end


Prefix notationPrefix notation

Obtain the prefix form of an expression when traverse its rooted tree in preorder.

Example.

By traversing it in preorder

receive the prefix notation for

this expression:

+ + x y 2 / – x 4 3

y

+

x

-

x 4

/

32

*

+



T1,T2,…Tn are the subtrees at r from left to right. The inorder traversal begins by traversing T1 in inorder, then visiting r. It continues by traversing T2 in inorder, then T3 in inorder, … , and finally Tn in inorder. Step 2: Visit r

T1 T2 Tn

Step 1: Visit T1

in inorder

Step 3: Visit T2

in inorder

Step n+1: Visit Tn

in inorder

r


The The inorderinorder traversal traversal

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Algorithm of Inorder TraversalAlgorithm of Inorder Traversal

Algorithm Inorder Traversalprocedure inorder(T: ordered rooted tree)r: = root of Tif r is a leaf then list relsebegin l: = first child of r from left to right T(l) : = subtree with l as its root Inorder (T (l)) list for each child c of r except for l from left to right T (c) : = subtree with c as its root Inorder (T (c))end


Infix notationInfix notation

Obtain the infix form of an expression when traverse its rooted tree in inorder.

Example.

By traversing it in inorder

receive the infix notation for

this expression:

x+y2+x–4/3x+y/2

y

+

x

-

x 4

/

32

*

+



T1,T2,…Tn are the subtrees at r from left to right. The

postorder traversal begins by traversing T1 in

postorder, then T2 in postorder,…, then Tn in

postorder, and ends by visiting r. Step n+1: Visit r

T1 T2 Tn

Step 1: Visit T1

in postorder

Step 2: Visit T2

in postorder

Step n: Visit Tn

in postorder

r


The pThe postoostorder traversalrder traversal

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Algorithm of Postorder TraversalAlgorithm of Postorder Traversal

Algorithm Postorder Traversal.procedure postorder (T: ordered rooted tree)r : = root of Tfor each child c of r from left to rightbegin T (c) = subtree with c as its root postorder(T(c))endlist r


Postfix notationPostfix notation

Obtain the postfix form of an expression when traverse its rooted tree in postorder .

Example.

By traversing it in postorder

receive the postfix notation for

this expression:

x y + 2 x 4 – 3 / +

y

+

x

-

x 4

/

32

*

+



Obtain the prefix form of an expression when traverse its rooted tree in preorder. Expressions written in prefix form are said to be in Polish notation (which is named after the logician Jan Lukasiewicz) .

Obtain the infix form of an expression when traverse its rooted tree in inorder.

Obtain the postfix form of an expression by traversing its binary tree in postorder. Expressions written in postfix form are said to be in reverse Polish notation.


A similar binary treesA similar binary trees

Two binary trees are called a similar if they have identical structure, e. m. they are either empty or contain identical number subtrees and their left and right subtrees are similar.


Equivalent binary treesEquivalent binary trees

Binary trees are equivalent, if they are similar and corresponding units contain the same information.

knure, software department, ph. 7021-446, e-mail: [email protected]

Documents

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parent of v

rooted tree

basic definitionsa tree

spainning tree h of

chord of spainning tree

knurethe trees

quantity of trees