Transcript Foundation of Computing Systems

Foundation of Computing Systems
Lecture 4
Trees: Part I
Tree: Example
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Tree: Definition
• A tree is a finite set of one or
more nodes such that:
• (i) there is a specially
designated node called the
root
• (ii) remaining nodes are
partitioned into n (n > 0)
disjoint sets T1, T2, . . ., Tn,
where each Ti (i = 1, 2, . . .,
n) is a tree; T1, T2, . . ., Tn
are called sub trees of the
root.
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T = (A(B(E, F(K, L))), C(G), D(H, I, J))
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Binary Tree
• A binary tree is a special form of a
tree
A
• A binary tree T is a finite set of
nodes, such that
• (i) T is empty (called empty binary
tree), or
• (ii) T contains a specially designated
node called the root of T, and the
remaining nodes of T form two
disjoint binary trees T1 and T2 which
are called left sub-tree and the right
sub-trees, respectively.
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Full Binary Tree
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Level
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Nodes
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A full binary tree of height 4
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Complete Binary Tree
Level Nodes
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A complete binary tree of height 4
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Skewed Binary Tree
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Properties of Binary Trees
• Property 1
• In any binary tree, maximum number of nodes on level l is 2l,
where l ≥ 0.
• Property 2
• Maximum number of nodes possible in a binary tree of height
h is 2h – 1.
• Property 3
• Minimum number of nodes possible in a binary tree of height
h is h.
• Property 4
• For any non-empty binary tree, if n is the number of nodes
and e is the number of edges, then n = e + 1.
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Properties of Binary Trees
• Property 5
• For any non-empty binary tree T, if n0 is the number of leaf
nodes (degree = 0) and n2 is the number of internal node
(degree = 2), then n0 = n2 + 1.
• Property 6
• Height of a complete binary tree with n number of nodes is
log 2 (n  1)
• Property 7
• Total number of binary tree possible with n nodes is
1 2n
Cn
n 1
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Representation of Binary Trees
• Linear representation
– Using array
• Linked representation
– Using linked list structure
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Linear Representation of Binary Trees
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The root node is at location 1.
2.
For any node with index i, 1 < i
£ n, (for some n):
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(a) PARENT(i) = i 2
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For the node when i = 1, there is no
parent.
(a) A binary tree
(b) LCHILD(i) = 2 * i
If 2 * i > n, then i has no left child.
(c) RCHILD(i) = 2 * i + 1
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If 2 * i + 1 > n, then i has no right
child.
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(b) Array representation of the binary tree
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Linked Representation of Binary Trees
DATA
Address
RCHILD DATA LCHILD
LC
RC
Node content
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(a) A binary tree
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Linked Representation of Binary
Trees
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(a) A binary tree
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(c) Logical view of the linked representation of a binary tree
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Binary Tree: Operations
• Major operations on a binary tree can be listed
as:
• Insertion. To include a node into an existing (may be empty)
binary tree.
• Deletion. To delete a node from a non-empty binary tree.
• Traversal. To visit all the nodes in a binary tree.
• Merge. To merge two binary trees into a larger one.
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Binary Tree: Insertion
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Binary Tree: Deletion
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Binary Tree: Traversals
• Depth-first search
R
• Breadth-first search
(level-by search)
Tl
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• Recursive search
1. R Tl
2. Tl R
3. Tl Tr
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Tr R Tl
R Tr Tl
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Binary Tree: Traversals
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Binary Tree: Traversals
R
Tl
R Tl Tr
Tl R Tr
Tl Tr R
Tr
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+-AB*C/EF
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(Preorder)
(Inorder)
(Postorder)
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F/E*C+B–A
AB–CEF/*+
+*/FEC–BA
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Binary Tree: Traversals
Preorder traversal
• Visit the root node R.
• Traverse the left sub-tree of R in preorder.
• Traverse the right sub-tree of R in preorder.
Inorder traversal
• Traverse the left sub-tree of the root node R is inorder.
• Visit the root node R.
• Traverse the right sub-tree of the root node R is inorder.
Postorder traversal
• Traverse the left sub-tree of the root R in postorder
• Traverse the right sub-tree of the root R in postorder
• Visit the root node R.
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Binary Tree: Merging
T1
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Different Binary Trees
• There are several types of binary trees possible each with its own
properties.
• Expression tree
• Binary search tree
• Heap tree
• Threaded binary tree
• Huffman tree
• Height balanced tree (also known as AVL tree)
• Red black tree
• Splay tree
• Decision tree
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