Binary Search Trees

CSE 331Section 2James Daly

Homework 1

• Homework 1 is due on today• Leave on the front table

Review: SampleTrees

…T1 T2 Tk

Review: Terminology

• Parent• Child• Ancestor• Descendent• Root• Leaf• Internal

A

B C

D

Review: Terminology

• Path• Depth• Height

A

B C

D

Review: Binary Tree

• Every node has at most 2 children• Left and right child

• Variation: n-ary trees have at most n children

Binary Search Tree

• For every node• Left descendents smaller (l ≤ k)• Right descendents bigger (r ≥ k)

k

<k >k

Traversal

• Three types• Pre-order

• Visit the parent before either of its children

• In-order• Visit the left children before the parent• Visit the right children after

• Post-order• Visit the children before the parent

Eval Tree

*

+ +

a b c -

d 3

Pre-order: * +ab +c -d3In-order: a + b*c + d – 3Post-order: ab+ cd3- + *

Search Tree

4

2 8

1 3 6 9

5 7

Search Tree

• All keys in left subtree <= root• All keys in right subtree >= root• In-order traversal => non-decreasing list

• Tree-sort• O(n log n) build time for balanced trees• O(n) time to traverse tree

• Can define functions to find particular items or the largest or smallest item

Find(t, x)

If (t = null)null

Else If (x < t.data)Find(t.left, x)

Else If (x > t.data)Find(t.right, x)

Elset

4

3

2

8

6 9

5 7

7?

FindMin(t) [Recursive]

If (t.left != null)t

ElseFindMin(t.left)

Insert(t, x)

If (t = null)t = new Node(x)

Else If (x < t.data)Insert(t.left, x)

Else If (x > t.data)Insert(t.right, x)

5

3 9

2

Construct a BST for 5, 3, 9, 2

Delete: 1st Case

• Leaf Node

6

5 7

6

5

Delete: 2nd Case

• One Child

6

7

88

7

Delete: 3rd Case

• Two children• Swap with least successor (or greatest

predecessor)• Then delete from the right (or left) subtree

Delete: 3rd Case

4

2 8

1 3 6 9

5 7

Delete: 3rd Case

5

2 8

1 3 6 9

4 7

Delete: 3rd Case

5

2 8

1 3 6 9

4 7

Next Time

• Balanced binary search trees• AVL trees• Maybe Red-Black trees