Lecture 14

Learning objectives

After this class, you should be able to:

1. Explain the following terms: tree, root, level, height, parent, child, ancestor, descendant, leaf, subtree, path, length of a path, binary tree, complete binary tree, and binary search tree.
2. Derive bounds on the heights of different types of trees.
3. Given a tree, show the order in which the vertices are visited with the following traversals: (i) preorder, (ii) postorder, (iii) inorder, and (iv) level-order.
4. Given a tree, determine whether or not it is a binary search tree.
5. Given a binary search tree, show the nodes visited while searching for an element.
6. Derive the time complexity of searching a binary search tree.
7. Write code to implement the four traversals mentioned above, and to search for an element in a binary search tree.

Reading assignment

1. Sections 4.3 (up to page 129), 4.6.
2. Lecture: Trees: Part 3 -- Binary search trees.
3. Pages 129-130.

Exercises and review questions

• Exercises and review questions on current lecture's material
  1. Draw a binary search tree that stores the following data in a tree of minimum possible height: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 2.5, 3.5, 8.5.
  2. For the tree you drew above, give the value of the root node, and the parent and children of the node with key 10.
  3. Show the nodes visited while searching the above tree for key 13.
  4. Give the order in which nodes are visited when you traverse the above tree using level-order traversal.
  5. Give the order in which nodes are visited when you traverse the above tree using inorder traversal.
  6. Give the order in which nodes are visited when you traverse the above tree using preorder traversal.
  7. Give the order in which nodes are visited when you traverse the above tree using postorder traversal.
  8. Which type of traversal on a binary search tree will lead to the elements being visited in ascending order?
  9. Write code to perform a level-order traversal of a binary search tree.
  10. Let a ternary tree be one in which each node can have up to three children. Derive a formula for the number of nodes of a complete ternary tree of height k. What is the minimum height of a ternary tree with n nodes? What is the maximum height of a ternary tree with n nodes?

• Questions on next lecture's material
  1. Let a binary tree consist of only one node, with value 15. If we wish to insert 3 and 18, then what will the tree look like after the insertions?