Binary%20Trees%20Terminology

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Binary TreesTerminology

• A graph G ltV,Egt is a collection of nodes and
  edges. An edge (v1,v2) is a pair of vertices that
  are directly connected.
• A path, P, is a sequence of vertices ltv1,v2,,vngt
  where and (vi,vi1) is an edge of G. A path
  ltv1,v2,,vn,v1gt is a cycle.
• A tree is a graph without cycles. The Root is a
  distinguished node of the tree. The Parent is the
  adjacent node that is closer to the root. A Leaf
  is a node without children. A child is an
  adjacent node that is further from the root. The
  level of a vertex is the length of the path from
  the root.
• A binary tree is a tree with at most two children

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Advantage of TreesAssuming Deletions exclude
time for Find

• Ordered Array
• Insert O(N), Delete O(N), Find O(Lg N).
• Unordered Array
• Insert O(1), Delete O(1), Find O(N).
• Linked List
• Insert O(1), Delete O(1), Find O(N).
• Binary Trees
• Insert O(1), Delete O(1), Find O(lg N) in most
  cases.
• Question When would Find not take place in O(Lg
  N)?

Note the above insert and removed complexities
dont include the find
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Tree StructureQuestion Can a Tree be
implemented using an array?

• typedef struct TreeNode
• Key key
• Data data
• struct TreeNode left
• struct TreeNode right
• Tree
• Top of the Tree Tree root
• Basic Tree Abstract Data Type Methods
• Node find(Key key)
• Item insert(Key key, Data data )
• Item delete(Key key)

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Binary Tree Find

• Node find( Key key)
• Node current root, previous
• previous null
• if (current null) return null
• while(!equal(current-gtkey, key))
• previous current if
  (greater(current-gtkey, key)) current
  current-gtleft
• else current current-gtright
• if (current null) return previous
• return previous

Notes -gt in C is equivalent to . in Java Java
has no equivalent to . in C
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Binary Tree Insertion/Deletionpseudo code

• Insert node
• Perform Find
• If node was found, Return false
• If tree was empty, Set root node
• Else If node lt leaf node returned by Find
• Link node to left child of
  leaf, Return true
• Else Link node to right child of leaf,
  Return true
• Delete node
• Perform Find
• If node was not found, Return false
• If node was a leaf, Unlink parent left or right
  pointer, Return true
• If node had one child, Link child to appropriate
  parent pointer, Return true
• Find and Remove successor
• Replace node with successor in the tree, Return
  true

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Other Search Tree Operations

• Find Minimum
• Go left until the leaf is found
• Find Maximum
• Go right until the leaf is found
• Predecessor
• Go left, and the proceed right until the leaf is
  found
• Successor
• Go right, and then proceed left until the leaf
  is found

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Binary Tree TraversalSee Tree Example in Text

• Preorder Traversal
• Recursively Call until leaf is found
• Recursive step
• Visit, Traversal(Left), Traversal(Right)
• Inorder (Traversal(left), Visit,
  Traversal(right))
• Postorder (Traversal(left), Traversal(right),
  Visit)