Foundation of Computing Systems

1
Foundation of Computing Systems

• Lecture 4
• Trees Part I

2
Tree Example
3
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 gt
  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.

T (A(B(E, F(K, L))), C(G), D(H, I, J))
4
Binary Tree

• A binary tree is a special form of a tree
• 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.

5
Full Binary Tree
6
Complete Binary Tree
7
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
• Property 7
• Total number of binary tree possible with n nodes
  is
• 
  
  

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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

• The root node is at location 1.
• For any node with index i, 1 lt i n, (for some
  n)
• (a) PARENT(i)
  
  
• For the node when i 1, there is no
  parent.
• (b) LCHILD(i) 2 i
  
  
• If 2 i gt n, then i has no left child.
• (c) RCHILD(i) 2 i 1
  
  
• If 2 i 1 gt n, then i has no right child.

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Linked Representation of Binary Trees
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Linked Representation of Binary Trees
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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
• Breadth-first search (level-by search)
• Recursive search

1. R Tl Tr 4. Tr Tl R 2. Tl R Tr 5. Tr R Tl
3. Tl Tr R 6. R Tr Tl
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Binary Tree Traversals
- A B C / E F
F E / C B A
A B C E / F
F / E C B A
A B C E F /
/ F E C B A
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Binary Tree Traversals
R Tl Tr (Preorder) Tl R Tr
(Inorder) Tl Tr R (Postorder)
- A B C / E F
F E / C B A
A B C E / F
F / E C B A
A B C E F /
/ F E C B A
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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.

21
Binary Tree Merging
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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