Topic 10 Trees.

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Presentation transcript:

Topic 10 Trees

Tree Definitions

Tree Definitions

Tree Definitions

Binary Search Tree

Traversal Operations tree

Announcements Participation 4 is up Homework 3 grading

Search

Insertion

Deletion Need Min/max Successor/predecessor

Pseudocode

Pseudocode

AVL

Height of AVL Tree

Proof

Insertion

Example

Single Right

Left-Right

Four cases

Deletions

Example

Announcements Test in 2 weeks Participation due tomorrow Clarification

Splay Trees

Amortized Running Time Runtime:

Basic Operations

When to Splay

Zig

Zig-Zag

Zig-zig

Example

Deletion

Optimal BST

Notation

Example

Example

More Notation (i,j) (j+1, j)

Observations

Observations

Announcements Participation due Clarifications on previous topics today Splay tree deletion Universal Hashing

Guaranteeing O(n) time

Birthday Paradox

Universal Hashing (clarification)

Computing the Optimal BST How to compute cost? Subtree at different depths

w(i,j)

AST

AST

Example Sequences/Probabilities

Leaves

The Big Idea

Subtrees with one key

Subtrees with 2 keys

Solution

Announcements Homework 5 due Homework 6 up Solutions will be up later today Homework 6 up

2-3 Trees A search tree Properties

Insertion

Deletion Remove Borrow Coalesce Apply recursively

Remove

Borrow

Coalesce

Generalizing/Example

2-3-4 trees, B-trees Why use 2-3 trees?