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Trees
Московский Государственный ниверситет
им. Циолковского
Студент : Заливнов Олег
Групп : МС-II-23
Лекция : 8
Тем : 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 dataа canа haveа different types (string,integer
and so on). But if to talk about complex data constructionа it have no type. Complex data constructions consist of simple
data, and CDC are stored as data searching algorithm. and that
is whyа CDCа areа the "selectors" - mechanism of searching and
accesing of data.
Such kindsа 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 asа itа describedа inа theа 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 аwordа inа bracketsа (e.g. (Oleg) means the
elementary data construction).
The most powerful way of description a CDC is a tree.
NOW WHAT IS TREE ?
Tree isа aа connectedа undirectedа graphа withа noа 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 isа notа theа treeа tooа becauseа thisа graphа 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 suchа treeа a
ROOTED tree.
Example 4 :
A ──── B A ─── B A ─── B
│ │
а│ │
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 uniqueа vertex A is called PARENT of vertex B if there
is a directed edge from A to B. When vertex Aа isа parentа of
vertex B, vertex B is called a CHILD of vertex A.
Vertices with the same parentа areа calledа SIBLINGS. The
ANCESTORS ofа a vertex other then th eroot are the vertices in
аthe path from root to this vertix, excluding the vertex itself
(that isа itsа parents, parents of its parents and so on...).
The DESCENDANTS of a vertex A are those vertices which haveа A
as an ancestor.
If a vertex of a tree has no children it is calle aа LEAF.
If a vertex has children it is called INTERNAL VERTEX.
If A is a vertex in a tree, the subgraph of a treeа which
consists ofа Aа 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 rootedа treeа isа called an M-ARY TREE if every internal
vertex has no more then M children. The tree is called a FULL
M-ARY treeа ifа everyа 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 vertexа in
the rootedа tree. Such trees are called ORDERED ROOTED TREES.
In such trees children are drawn in order from left to right.
In anа ordered binary tree, if an internal vertex has two
children, first is called LEFT CHILD, second is calledа RIGHT
CHILD.
If a subtree has a left child of a vertex as aа rootа then
such subtree is called LEFT SUBTREE OF A VERTEX. If a root of
a subtree is a right child ofа aа vertexа thenа weа callа such
subtree RIGHT SUBTREE OF A VERTEX.
We willа 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 HEIGHTа ofа 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. аI'llа justа name
them :
1) Anа 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) Aа fullа m-aryа treeа with
(a) n verticesа hasа iа =а (n-1)/mа 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) Thereа areа atа mostа m^hа leavesа in any m-ary tree of
height = h.
There are several ways of drawing a tree.
First one to draw a trer as aа diagramа wasа presentedа in
previous examples, but there are some more ways to do it.
Second wayа of representing a tree is a brackets
representation. Inа thisа wayа theа internalа brackets present
sub-trees.
Example 8 : (C is a root)
D ─── C ─── G
│ │ ====== (C,(D,F,G,(H)))
F H
The thirdа 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 isа oneа veryа importantа applicationа ofа treesа in
encoding systems.
The task of encoding system is to enter codes of wordsа or
frase soа that message could be recoded. The main requirement
is the ability to synonymously restore the original textа with
the help of codes.
So for exampleа weа haveа a аbinaryа messageа andа aа 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 meansа thatа inа such
vocabularu no one word begins from another.
So our task is to divide message into symbolsа andа encode
them.
Example 10 :
We have the message : 11001
and the prefix full vocabulary : 1 E
01 L
001 G
O
And so this message can be divided into four symbols :
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 binaryа treeа representsа a
full,prefix coding vocabulary.
So in such way trees are used in encoding systems.