Trees
• Tree – is a connected undirected graph with no simple
circuits.
• Forest – is a graph with each its connected
component being a tree.
• Theorem( unique simple path )
A undirected graph is a tree  there is a unique simple
path between any two of its vertices.
• A rooted tree is a tree in which one vertex has been
designated as the root and every edge is directed away from
the root.
Terminologies: parent-child, leaf-internal node, siblings,
descendants-ancestors
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M-ary Tree (1/4)
• A rooted tree is called a m-ary tree if every internal
vertex has no more than m children.
• Applications：檔案系統架構、平行電腦架構
• Property：
1. A tree with N vertices has N-1 edges.
2. A full m-ary tree with i internal vertices contains
n = m * i +1 vertices.
Conversely, n vertices has i = ( n -1 ) / m internal vertices.
and L = [ ( m – 1 ) * n + 1 ] / m leaves.
Here “full” means each internal vertex has m children
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M-ary Tree (2/4)
• Theorem : A full m-ary tree with
<1> n vertices has i = ( n – 1 ) / m internal vertices
and L = [ ( m – 1 ) * n + 1 ] / m leaves.
<2> i internal nodes has n = m * i + 1 vertices
and L = ( m – 1 ) * i + 1 leaves.
<3> L leaves has n = ( m * L – 1 ) / ( m – 1 ) vertices
and i = ( L – 1 ) / ( m – 1 ) internal vertices.
Note: n = m*i + 1 & L = n-I =(m – 1)*i + 1
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M-ary Tree (3/4)


E.g. Suppose someone starts a chain letter. Each
person receives the letter is asked to send it on to 4
people. Some do it, some doesn’t send the letter.
How many people have seen the letter, if no one has
received more than one letter and the chain letter
ends after having 100 people who read it but did not
send it out ?
Answer:
 4-ary tree
 L = 100, internal nodes = (L – 1) / ( 4 – 1 )=33
n = 4 * 33 + 1 = 133
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M-ary Tree (4/4)
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
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The level of a vertex in a rooted tree = the
length of the unique path from the root to the
vertex.
Level of root = 0.
A rooted tree is of height h if the max of levels
of vertices = h.
A rooted m-ary tree of height h is balanced if
all leaves are at levels h or ( h – 1 ), h is the
height of the tree.
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Application of Trees (1/2)


Binary Search Tree
 左小右大，alphabetical ordering
E.g. Math, Geography, Meteorology, Chemistry, Physics,
Zoology, Geology, Psychology, Oceanography
Prefix Codes

bit string of a letter never occurs as the 1st part of another letter.
e:0
a : 10
t : 110
n : 1110
s : 1111
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Application of Trees (2/2)

Huffman Codes :
 Prefix output code so that frequently used letter
has fewer bit strings.
 設A,B,C,D,E,F當coding symbols，其出現頻率依序
為0.08, 0.10, 0.12, 0.15, 0.20, 0.35
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Tree Traversal


Universal address system of the ordered
rooted tree. (Revision Control System, OIDs )
 root is labeled by 0, its children is labeled
from left to right with 1,2,3,…
 For each vertex v at level n with label A,
label its children from left to right with A.1,
A.2, A.3, ...
Traversal Algorithms:
 Preorder, Inorder, Postorder
中左右 左中右 左右中
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Traversal Algorithms


PreorderabejknopfcdglmhI
Postorderjnopkefbclmghida
Let T be an ordered rooted tree.
If T consists only of r, then r is the preorder traversal of T.
Otherwise, suppose that 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.
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Inorder Traversal

Inorder jenkopbfaclgmdhi
Let T be an ordered rooted tree.
If T consists only of r, then r is the inorder traversal of T.
Otherwise, suppose that T1, T2, …, Tn are the subtrees at r from left
to right in T.
The inorder traversal begins by traversing T1 in inorder, then
visiting r. It continues traversing T2 in inorder and so on until
Tn is traversed in inorder.
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Infix, Prefix and Postfix (1/2)

((x+y)^2)+((x-4)/3)

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Inorder traversal of this
binary tree infix form
Infix form requires fully
parentheses.
Prefix traversal called
Polish Notation
Binary tree representation using prefix form
+^+xy2/-x43
(1) 從右到左 (2) 每遇到operator時執行運算
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Infix, Prefix and Postfix (2/2)



Preorder- + ^ + x y 2 / – x 4 3
從右到左，取operator右邊的operands.
Postorder- x y + 2 ^ x 4 – 3 / +
從左到右，取operator左邊的operands.
Inorder traversals of
all have the same outputs as : x + y / x + 3
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Spanning Tree


Let G be a simple graph.
A Spanning tree of G is a subgraph of G
that is a tree containing every vertex in G.
Theorem
A simple graph is connected  if has a
spanning tree.
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Depth-First-Search (DFS) &
Breadth-First-Search (BFS)

產生Spanning tree的方法：
1. Removing edges from simple circuits.
2. Adding edges from ( an arbitrary vertex )
either Depth-First or Breadth-First.
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BFS example
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Minimum Spanning Tree


Prim’s algorithm:先選一minimum
edge, 再加入鄰近cost較小之
edge (不造成circle)
Kruskal’s algorithm:逐一加入
minimum edge, 只要不造成
circle即可
2010/9/23
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Prim’s
Kruskal’s
b,f
c,d
a,b
k,l
f,i
b,f
a,e
c,g
i,j
a,b
f,g
f,j
c,g
b,c
c,d
j,k
g,h
g,h
h,l
i,j
k,l
a,e
16
Kruskal’s Algorithm

Procedure Kruskal (G: weighted connected
undirected graph with n vertices)
T := an empty graph
for i:= 1 to n-1
begin
e = an edge in G with smallest weight that does
not form a simple circuit in T if added to T.
T := T with e added
end
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Prim’s Algorithm

Procedure Prim (G: weighted connected
undirected graph with n vertices)
T := a min-weight edge
for i:= 1 to n-2
begin
e = an edge, not in T, of min weight and not
forming a simple circuit in T if added to T.
T := T with e added
end
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MEMO



Read chapter 10 in details.
Get familiar with the relationship between the
number of edges and vertices in m-ary tree,
tree traversals, prefix codes and Huffman codes,
and spanning tree.
HW #17,19-21 of §10.1, #1,13,14,19,21,23,24
of §10.2, #17,18,22,23 of §10.3, # 16,17 of
§10.4, #3,4,7,8 of §10.5
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