Trees
Московский Государственный Университет
им. Циолковского
Студент : Заливнов Олег
Группа : 5МС-II-23
Лекция : 8
Тема : Деревья
TREES
Plan:
1) The tree presenation of data constructions.
2) What is tree?
a) definition
b) the terminology
c) types of trees
3) Tree applications in encoding systems.
Elementar data can have different types (string,integer
and so on). But if to talk about complex data construction -
it have no type. Complex data constructions consist of simple
data, and CDC are stored as data searching algorithm. and that
is why CDC are the "selectors" - mechanism of searching and
accesing of data.
Such kinds of data as complex data constructions are need
to organize search.
We can describe CDC in different ways. For example we can
describe it in the way as it described in the programming
language Cobol :
1 University
2 (first fac.)
2 (second fac.)
2 (third fac.)
2 (fourth fac.)
2 fifth fac.
3 PM
4 (Pasha)
4 (Andrey)
3 IT
4 (Zhenia)
4 (Olga)
3 MS
4 (Oleg)
4 (Helen)
4 (Artem).
Where the word in brackets (e.g. (Oleg) means the
elementary data construction).
The most powerful way of description a CDC is a tree.
NOW WHAT IS TREE ?
Tree is a connected undirected graph with no simple
circuits. So a tree cannot contain multile edges or loops, and
so tree is a simple graph.
Example 1 :
D ─────────── A ──────────── C
│ │ │
│ │ │
│ B ──── F │
│ │
E H ──── G ───── I ───── J
this is a tree ;
Example 2 :
E ────────── A ────────── B
│ │ │
│ │ │
F D─────────── C
it is not the tree, because path A-B-C-D is a loop;
Example 3 :
A ─────── B
│
D ────┼──── E ────── F
│
C
it is not the tree too because this graph is not
connected;
Also we can select a special vertex and call it a root and
assign the direction to each edge. And we call such tree a
ROOTED tree.
Example 4 :
A ──── B A ─── B A ─── B
│ │
│ │
D ──── C ──── G D ─── C ─── G D ─── C ─── G
│ │ │ │ │ │
│ │
F H F H F H
a)Unrooted tree . b) Rooted tree c) Rooted tree
with root A . with root C .
The unique vertex A is called PARENT of vertex B if there
is a directed edge from A to B. When vertex A is parent of
vertex B, vertex B is called a CHILD of vertex A.
Vertices with the same parent are called SIBLINGS. The
ANCESTORS of a vertex other then th eroot are the vertices in
the path from root to this vertix, excluding the vertex itself
(that is its parents, parents of its parents and so on...).
The DESCENDANTS of a vertex A are those vertices which have A
as an ancestor.
If a vertex of a tree has no children it is calle a LEAF.
If a vertex has children it is called INTERNAL VERTEX.
If A is a vertex in a tree, the subgraph of a tree which
consists of A and all its descendants and all edges incident
to these descendants is called a SUBTREE with a root A.
Example 5 :
A ─── B
│
D ─── C ─── G D ─── C ─── G
│ │ │ │
F H F H
(a) Tree T (b) Subtree T1
A - is a root
A - is a parent of B and C.
C - is a child of A
C and B - are siblings
C - is an ancestor of H
H - is an descendant of A
F - is a leaf
C - is an internal vertex
A rooted tree is called an M-ARY TREE if every internal
vertex has no more then M children. The tree is called a FULL
M-ARY tree if every internal vertex has exactly M children.
And if M = 2 then such M-ary tree is called BINARY TREE.
Example 6 :
A ─── B A
│
│
D ─── C ─── G D ─── C ─── G ─── E
│ │ │ │
F H F H
a) 3-ary tree b) full 3-ary tree
with root A. with root C.
C ─── G ── B
│ │
F H
c) binary tree
with root C.
Also we can order the children of each internal vertex in
the rooted tree. Such trees are called ORDERED ROOTED TREES.
In such trees children are drawn in order from left to right.
In an ordered binary tree, if an internal vertex has two
children, first is called LEFT CHILD, second is called RIGHT
CHILD.
If a subtree has a left child of a vertex as a root then
such subtree is called LEFT SUBTREE OF A VERTEX. If a root of
a subtree is a right child of a vertex then we call such
subtree RIGHT SUBTREE OF A VERTEX.
We will call the LEVEL of a vertex V in a rooted tree the
length of the unique path from the root to the vertex V.
The level of root equal 0.
The HEIGHT of a rooted tree is the length of its longest
path from the root to any vertex.
Example 7 :
D ─── C ─── G
│ │
F H
The root is vertex C.
The level of F is 1.
The height of the tree is 2.
There are several theoremes about trees. I'll just name
them :
1) An undefined graph is a tree if and only if there is a
unique simple path between any two 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.
4) A full m-ary tree with
(a) n vertices has i = (n-1)/m internal vertices
and l = [(m - 1)n + 1]/m leaves.
(b) i internal vertices has n = mi+1 vertices and
l = (m-1)i + 1 leaves.
(c) l leaves has n=(ml-1)/(m-1) vertices and
i = (l-1)/(m-1) internal vertices.
5) There are at most m^h leaves in any m-ary tree of
height = h.
There are several ways of drawing a tree.
First one to draw a trer as a diagram was presented in
previous examples, but there are some more ways to do it.
Second way of representing a tree is a brackets
representation. In this way the internal brackets present
sub-trees.
Example 8 : (C is a root)
D ─── C ─── G
│ │ ====== (C,(D,F,G,(H)))
F H
The third way is to present tree as a consistent numbered
sections.
Example 9 :
D ─── C ─── G 1. C
│ │ ========== 1.1. D
1.2. F
F H 1.3. G
1.3.1. H
All the ways of presenting trees are equalent.
There is one very important application of trees in
encoding systems.
The task of encoding system is to enter codes of words or
frase so that message could be recoded. The main requirement
is the ability to synonymously restore the original text with
the help of codes.
So for example we have a binary message and a code
vocabulary. I must say that not every vocabulary can be a code
vacabulary. The requirements to it are the following :
1) it must be full
2) it must be prefix vocabulary, it means that in such
vocabularu no one word begins from another.
So our task is to divide message into symbols and encode
them.
Example 10 :
We have the message : 000011001
and the prefix full vocabulary : 1 E
01 L
001 G
000 O
And so this message can be divided into four symbols :
000 01 1 001
and then can be encoded as OLEG.
It is not difficult to mention that this vocabulary can be
presented as a binary tree.
Then we can mention that every binary tree represents a
full,prefix coding vocabulary.
So in such way trees are used in encoding systems.