1. COMP 103 Binary Search Trees II Marcus Frean 2014-T2 Lecture 26
Lindsay Groves, Marcus Frean, Peter Andreae, and Thomas Kuehne, VUW
Marcus Frean
School of Engineering and Computer Science, Victoria University of Wellington
2014-T2 Lecture 26

2. RECAP-TODAY RECAP Tree traversals TODAY Binary Search Trees
Using hierarchical access to unstructured data
Reading: Chapter 17.1

3. Binary Search Again Searching “50” 20 5 52 1 13 50 99 1 5 13 20 50 52
mid
low hi

4. “≤” if duplicates are allowed
Binary Search Trees
“≤” if duplicates are allowed
Properties
For every node:
all items in left subtree < item
item < all items in right subtree
Ascending order obtained by ___________ traversal 20 5 52 1 13 50 99 51 8

5. Efficiency How many steps do we need to find an element?
Inverse question is easier to answer:
How many nodes fit into a binary tree of height n?
Answer: n = 2(height+1) -1
 Steps needed = height+1 = log2(n+1)
Height
Nodes 1 1 3 2 7 3 15

6. Efficiency
Does it help?
Number of Elements
Steps

7. This is why ensureCapacity() (cf. slide 17 in L6)
Efficiency
This is why ensureCapacity() (cf. slide 17 in L6)
doubles the array size!
Does it help?
Steps required
Elements

8. Efficiency Complexity of finding an element =
Complexity of adding an element =
Complexity of removing an element =
Fred
Celia
Joe
Anton
Des
Hans
Kim
Berta
Zoe

9. Efficiency Height depends on order of adding elements
 optimal case: O(log(n)) H G L B E J N A C D F I K M O H G L B E J N A C D F I K M O

10. Efficiency Height depends on order of adding elements
 worst case: O(n) A B C D E F G H I J K L M O P A B C D E F …

11. Balanced and Unbalanced BSTsH G B E J N L C F D K I O M A A H I L D F B K J N M C E G O H G B F M K N C A E D O I L J
see the discussion of rotations, in the text book

12. Efficiency Unbalanced tree: Balanced tree: worst case cost =
average case cost =
Balanced tree:
cf. “Random Binary Trees”
cf. “AVL”, “2-3”, “Red-Black” trees, etc.