Tutorial 7: Binary Search Tree

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Tutorial 7 Binary Search Tree

• Short review on binary tree
• Tree traversals
• Binary Search Tree (BST)
• Properties
• FindKey
• FindMin/FindMax
• Insert/Delete
• For sorting?

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Short review on binary tree

• At most two children for each node
• A root node at the top (or the tree is empty)
• Nodes with no childleaf nodes
• Left complete tree if
• All levels are full except last level
• Last level filled from left to right

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Tree traversals

• To visit each node of the tree
• Recursively visiting left and right sub-trees
• Pre-order (NLR)
• Current Node, Left sub-tree, Right sub-tree
• In-order (LNR)
• Left, Node, Right
• Post-order (LRN)
• Left, Right, Node

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Tree traversalsExamples

• Pre-order (NLR)
• 9,4,2,3,8,6,7
• Always print the node first
• In-order (LNR)
• 2,4,3,9,6,8,7
• Print whenever return from left
• Post-order (LRN)
• 2,3,4,6,7,8,9
• Print whenever return from right

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Tree traversalsQuestion

• Question Draw the binary tree whose tree
  traversal sequences are A, B, D, C, E in
  Preorder and B, D, A, C, E in Inorder.
• 1st element of preorder A root!
• Note given the root, inorder (LNR) has the left
  and right partitions B,D,A,C,E

A
B
D
E
C
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Tree traversals

• Preorder A, B, D, C, E and Inorder B, D, A, C,
  E
• Next level on the left A-gtB-gtD or A-gtD-gtB?
• Preorder (NLR), (i.e. A-gtLeft) give B first, so
  the levels on the left is A-gtB-gtD
• D is left or right child of B?
• Inorder (LNR), (i.e. Left-gtB-gtright) gives
  empty-gtB-gtD. So, D on the right

A
B
D
D
A
B
E
C
D
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Tree traversals

• Preorder A, B, D, C, E and Inorder B, D, A, C,
  E
• Next level on the right A-gtC-gtE or A-gtE-gtC?
• Preorder (NLR), (i.e. A-gtright) give C first, so
  A-gtC-gtE
• E is left or right child of C?
• Inorder (LNR), (i.e. Left-gtC-gtright) gives
  empty-gtC-gtE. So, E on the right

A
B
C
D
E
E
A
B
C
D
E
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Binary Search TreeProperties

• A Binary Tree (nodes at most two children)
• Left sub-tree lt Node lt Right sub-tree
• Which one(s) is/are BST(s)?

(1)
(2)
(3)
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Binary Search Tree

• Which one(s) is/are BST(s)?

(1)
(2)
(3)
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Binary Search Tree

• Which one(s) is/are BST(s)?

(3)
(1)
(2)
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Binary Search TreeSummary

• BST Left lt Node lt Right
• Heap Only requiring Parent gt Children
• WHOLE left (or right) sub-tree ltX (or gtX)
• All items in the sub-tree has to be ltX (or gtX)
• NOT necessarily a Complete tree!
• Worse case can be a linked list!
• Same collection of elements can have different
  BSTs

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Tree depth/height

• Depth of a node no. of edges from root
• Root is also a node, depth of root ?
• Height of a node length of longest path to a
  leaf
• Height of leaves ?
• Height of root height of tree

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Binary search treeFind key

• To look for a node with a certain Key
• Left lt Node lt Right
• Recursively do
• Compare(key,node)
• Keyltnode
• Go left
• Keygtnode
• Go right

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Binary search treeFind key

• Find the key 7

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Binary search treeFind the key 3

• node 6, Key lt 6
• Go left, node 2, Key gt 2
• Go right, node 4, Key lt 4
• Go left, node 3, Key 3
• O(1) at each level
• Find a key
• O(depth) if found
• O(height) if not found
• Time complexityO(height)

lt
gt
lt
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Binary search treeFind key

• search_binary_tree(node, key)
• if node is null
• return None // not found
• if key lt node.key
• return search_binary_tree(node.left, key)
• else if key gt node.key
• return search_binary_tree(node.right, key)
• else
• return node.value // are you sure this is ?

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Binary search treeFindMin and Find Max

• Left lt Node lt Right
• Min Left most node
• Recursively go left
• Until no more left child
• Max Right most node
• Recursively go right
• Until no more right child

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Binary search treeFindMin and Find Max

• find_min(node)
• if node is null
• return null // what is this?
• if node.left is null
• return node
• else
• return find_min(node.left)
• Really need recursion? Write the same function by
  iteration.

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Binary search treeFindMin and Find Max

• find_max(node)
• if node is not null
• while node.right is not null do
• node node.right
• return node
• Iterative version looks simpler

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Binary search treeInsert

• To insert a key into BST, similar to Find Key
• Proceed as if you want to find the key
• If found, duplicate key
• do nothing or update a counter or
• insert to a list or
• Otherwise insert X at the last spot.

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Binary search treeInsert the key 5

• node 6, Key lt 6
• Go left, node 2, Key gt 2
• Go right, node 4, Key gt 4
• Go right, NULL, insert 5
• O(1) at each level
• Again, O(height)

lt
gt
5
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Binary search treeInsert key

• insert(node, key)
• if node is null
• node new Node(key)
• else if key lt node.key insert(node.left, key)
• else if key gt node.key insert(node.right, key)
• else node.count // duplicate
• Question Can we do iterative insert?

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Binary search treeDelete

• To delete a key from a BST
• Proceed as if you want to find the key
• If found the node n, delete it!
• Is it really that simple?
• I wish it was
• Step two is a bit harder three cases

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Binary search treeDelete the key 1

• node 6, Key lt 6
• Go left, node 2, Key lt 2
• Go left, node 1, Key 1
• Case one no child
• Delete it!

lt
lt
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Binary search treeDelete the key 8

• node 6, Key gt 6
• Go right, node 8, Key 8
• Case two one child
• My parent bypass me, point to my child instead
• Really no violation?
• Left lt X lt Right

gt
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Binary search treeDelete the key 2

• node 6, Key lt 6
• Go left, node 2, Key 2
• Case two two child
• Replace node 2 with Minnode (m) of right
  sub-tree
• Recursively delete m
• Why choose this way?

lt
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Binary search treeDelete the key 2, before and
after
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Binary search treeDelete key

• delete(node, key)
• if node is null return // nothing to delete
• if key lt node.key delete(node.left, key)
• else if key gt node.key delete(node.right, key)
• else if both node.left and node.right are not
  null
• node.key find_min(node.right).key
• delete(node.right, node.key)
• else // no child or one child case
• oldNode node
• if node.left is not null nodenode.left
• else nodenode.right
• delete oldNode

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Binary search treeDelete Summary

• Case 1 node n is a leaf (no child) delete it!
• Case 2 node n has only one child
• Parent bypass n, point to n.child and delete n
• Case 3 node n has two child
• Replace n with smallest node (m) of the right
  sub-tree
• Recursively delete m
• O(height)?

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Time complexity

• All operations discussed takes O(height) time
• Average height O( log N ) (WHY?)
• Textbook pages 133-134
• Worse case (linked list) height ? O(N)