Binary Search Trees (8.4)

















What's a Binary Search Tree?


What's the average-case time for finding an item in a List?

	O(n)

What's the average-case time for finding an item in a BST?

	O(log n)


How many steps are needed to find an item in a 1,000,000-item List?

	an average of 500,000 steps

How many steps are needed to find an item in a 1,000,000-item BST?

	an average of 20 steps

















The BST Order Property


What's the defining property of a Binary Search Tree?

	for every node in the tree
	  items in the left subtree are smaller
	  items in the right subtree are larger


Draw an example BST on the board.

	     5
	    / \
	   3   6
	  / \   \
	 1   4   9
	  \     /
	   2   7
	        \
		 8

















Classwork
You may work with a partner.

Is the tree a Binary Search Tree?
If not, identify where the BST order property is violated.

	     5
	    / \
	   3   8
	  / \   \
	 1   4   9
	  \   \
	   2   6

	     5
	    / \
	   4   8
	  /   / \
	 2   6   9
	      \
	       7

















Searching


How do you find an item in a Binary Search Tree?

	compare the item to the root
	if smaller, repeat in the left subtree
	if larger, repeat in the right subtree


Show the steps of finding 7 in the BST.
Show the steps of finding 3 in the BST.

	     5
	    / \
	   4   8
	  /   / \
	 2   6   9
	      \
	       7

















How do you find the smallest or largest item in a Binary Search Tree?


Show the steps of finding the smallest item in the BST.
Show the steps of finding the largest item in the BST.


	     5
	    / \
	   3   6
	  / \   \
	 1   4   9
	  \     /
	   2   7
	        \
		 8

















Inserting


How do you insert an item into a Binary Search Tree?


Show the result of inserting 3 into the BST.

	     5
	    / \
	   4   8
	  /   / \
	 2   6   9
	      \
	       7

















Classwork
You may work with a partner.

Show the result of inserting 7 into the BST.

	     5
	    / \
	   3   8
	      / \
	     6   9


Show the result of inserting 4 into the BST.

	      7
	     / \
	    5   8
	   / \   \
	  2   6   9
	 / \
	1   3

















Removing


How do you remove an item from a Binary Search Tree?


Show the result of removing 3 from the BST.

	      7
	     / \
	    5   8
	   / \   \
	  2   6   9
	 / \
	1   3


Show the result of removing 9 from the BST.

	     5
	    / \
	   3   6
	  / \   \
	 1   4   9
	  \     /
	   2   7
	        \
		 8


Show the result of removing 2 from the BST.

	      7
	     / \
	    2   8
	   / \   \
	  1   5   9
	     / \
	    3   6
	     \
	      4

















Classwork
You may work with a partner.

Show the result of removing 5 from the BST.

	      6
	     / \
	    5   8
	   /   / \
	  2   7   9
	 / \
	1   4
	   /
	  3


Show the result of removing 3 from the BST.

	      3
	     / \
	    2   8
	   /   / \
	  1   5   9
	     / \
	    4   6
	    	 \
	          7

















Classwork
You may work with a partner.

Show the result of inserting  4 9 5 1 3 2 7 8 6  into a BST.

Show the result of removing  4 5 6  from the BST.

















BST Implementation


Write the code for insert.


Classwork
You may work with a partner.
Write the code for find.


Write the code for remove.

















BST Time Analysis


What's the time for find, insert, and remove on a BST?

What characteristic of the tree affects the time?

What's the height of a BST with N nodes?


What's the best-case height of a BST with N nodes?

	the tree is balanced
	the height is O(log N)
	find, insert, and remove are O(log N)


What's the worst-case height of a BST with N nodes?

	the tree is like a linked list
	the height is O(N)
	find, insert, and remove are O(N)


What's the average-case height of a BST with N nodes?

	the average case is 38% worse than the best case
	the height is still O(log N)
	find, insert, and remove are O(log N)


What happens if you insert a sorted sequence of items into a BST?