11 Binary Tree Data Structures

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11Binary Tree Data Structures

• Binary trees and binary search trees.
• Searching.
• Traversal.
• Implementation of sets using BSTs.

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BST search (1)

• Problem Search for a given target value in a
  BST.
• Idea Compare the target value with the element
  in the root node.
• If the target value is equal, the search is
  successful.
• If target value is less, search the left subtree.
• If target value is greater, search the right
  subtree.
• If the subtree is empty, the search is
  unsuccessful.

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BST search (2)

• BST search algorithm
• To find which if any node of a BST contains an
  element equal to target
• 1. Set curr to the BSTs root.2. Repeat 2.1. I
  f curr is null 2.1.1. Terminate with answer
  none. 2.2. Otherwise, if target is equal to
  currs element 2.2.1. Terminate with answer
  curr. 2.3. Otherwise, if target is less than
  currs element 2.3.1. Set curr to currs left
  child. 2.4. Otherwise, if target is greater than
  currs element 2.4.1. Set curr to currs right
  child.

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BST search (3)

• Animation (successful search)

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BST search (4)

• Animation (unsuccessful search)

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BST search (5)

• Analysis (counting comparisons)
• Let the BSTs size be n.
• If the BST has depth d, the number of
  comparisons is at most d 1.
• If the BST is well-balanced, its depth is
  floor(log2 n)
• Max. no. of comparisons floor(log2 n) 1
• Best-case time complexity is O(log n).
• If the BST is ill-balanced, its depth is at most
  n1
• Max. no. of comparisons n
• Worst-case time complexity is O(n).

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BST search (6)

• Implementation as a Java method (in class BST)
• public BSTNode search (Comparable target)
• int direction 0 BSTNode curr
  root for ()
• if (curr null) return
  null direction target.compareTo(curr.element)
  if (direction 0) return curr else if
  (direction lt 0) curr curr.left else curr
  curr.right

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Binary tree traversal

• Binary tree traversal Visit all nodes (elements)
  of the tree in some predetermined order.
• We must visit the root node, traverse the left
  subtree, and traverse the right subtree. But in
  which order?
• In-order traversal Traverse the left subtree,
  then visit the root node, then traverse the right
  subtree.
• Pre-order traversal Visit the root node, then
  traverse the left subtree, then traverse the
  right subtree.
• Post-order traversal Traverse the left subtree,
  then traverse the right subtree, then visit the
  root node.

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Binary tree in-order traversal (1)

• Schematic for in-order traversal

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Binary tree in-order traversal (2)

• Illustration

• In-order traversal of a BST visits the elements
  in ascending order.

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Binary tree in-order traversal (3)

• Binary tree in-order traversal algorithm
  (generic)
• To traverse, in in-order, the subtree whose
  topmost node is top
• 1. If top is not null 1.1. Traverse, in
  in-order, tops left subtree. 1.2. Visit
  top. 1.3. Traverse, in in-order, tops right
  subtree.2. Terminate.
• This algorithm is generic the meaning of Visit
  is left open.

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Example printing elements in order

• Java method
• public static void printInOrder (BSTNode top)
  // Print, in ascending order, all the elements
  in the BST subtree // whose topmost node is
  top. if (top ! null)
• printInOrder(top.left) System.out.print
  ln(top.element) printInOrder(top.right)

Visit top (printing its element).
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Binary tree pre-order traversal (1)

• Schematic for pre-order traversal

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Binary tree pre-order traversal (2)

• Illustration

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Binary tree pre-order traversal (3)

• Binary tree pre-order traversal algorithm
  (generic)
• To traverse, in pre-order, the subtree whose
  topmost node is top
• 1. If top is not null 1.1. Visit
  top. 1.2. Traverse, in pre-order, tops left
  subtree. 1.3. Traverse, in pre-order, tops
  right subtree.2. Terminate.

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Binary tree post-order traversal (1)

• Schematic for post-order traversal

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Binary tree post-order traversal (2)

• Illustration

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Binary tree post-order traversal (3)

• Binary tree post-order traversal algorithm
  (generic)
• To traverse, in post-order, the subtree whose
  topmost node is top
• 1. If top is not null 1.1. Traverse, in
  post-order, tops left subtree. 1.2. Traverse,
  in post-order, tops right subtree. 1.3. Visit
  top.2. Terminate.

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Implementation of sets using BSTs (1)

• Represent an (unbounded) set by a BST whose
  elements are the set members.

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Implementation of sets using BSTs (2)

• The BST representation of a set is not unique

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Implementation of sets using BSTs (3)

• Summary of algorithms and time complexities