CSC 301.01, Class 11: Binary search trees

Overview

• Preliminaries
  • Notes and news
  • Upcoming work
  • Extra credit
  • Questions
• Tree basics & terminology
• Trees, formalized
• Binary trees
• Traversing trees
• Binary search trees
• Building balanced BSTs.
• Balancing BSTs.

News / Etc.

• As seems to be the norm, my weekend failed to produce as much work time as I needed, and work took longer. I remain behind on grading.
• We’ll be discussing Red/Black trees on Wednesday and Friday.

Upcoming work

• Assignment 4, due 10:30 p.m. Wednesday
  • Implement hash tables in Scheme.
  • Reflect on how to implement sets.
• Exam 1 to be distributed Wednesday.

Extra credit (Academic)

• CS Table, Tuesday, Hacktivism
• CS Extras, Thursday, 4:15 p.m., 3821, Study Abroad in CS
• One of the Happy Kareoke Fun Time activities
  • Musical, Friday 9pm, Gardner.
  • Workshop: Burn your fear. Saturday, 2pm. The Wall. Email [improv] to sign up.
  • Workshop: Creating through improv. Sunday, 2pm. The Wall. Email [improv] to sign up.
  • Musical, Sunday, 7pm, Gardner

Extra credit (Peer)

Extra Credit (Misc)

Questions

How do I expand the hash table?

(define hash-table-new
  (lambda ()
    (vector 'hash-table (make-vector 100 null) ...)))

(define hash-table-expand
  (lambda (table)
    (let ([new-contents (make-vector (+ 1 (* 2 (hash-table-size table))))])
      ... ; copy the elements
      (vector-set! table 1 new-contents))))

Are there limitations to what can be in the key?

Any ASCII strings are valid. But you have char->integer in Scheme.

So there are 128 different characters.

Remember: The hash function should give you a number between 0 and infinity. You decide where to place it by mod’ing by the hash table size.

Trees, introduced

Talk with your group about the following questions.

Is “Tree” an ADT or a Data Structure? Both? Neither?

• ADT: Because data structures are things you directly implement.
• Data structure: There is a natural implementation: Nodes with pointers.

Detour: List is an ADT (we are organizing elements for iteration and support the following operations: …). LinkedList or ArrayList is a Data Structure (each element has a link to the next; the elements are stored in an array)

It’s complex. A Tree is a way of organizing info, which may be more abstract than ADT. But we think of trees as “Nodes with pointers” which makes them pretty concrete. And we use them to implement other ADTs (e.g., “We implement dictionaries with binary search trees.”)

Note: We also represent binary trees as arrays with the understanding that the children of the value at position i are at positions 2i+1 and 2i+2. What are the central key operations?

Trees, formalized

Basic definition

We will denote “a tree of values of type X” as Tree.

• If x is an X, Leaf(x) is a Tree
• If x is an X and T1 … Tn are all Trees, then so is Node(x,T1,T2,...,Tn).

Empty leaves

We will denote “a tree of values of type X” as Tree.

• Leaf is a Tree
• If x is an X and T1 … Tn are all Trees, then so is Node(x,T1,T2,...,Tn).

Values only at leaves

• If x is an X, Leaf(x) is a Tree
• If T1 … Tn are all Trees, then so is Node(T1,T2,...,Tn).

Terminology

Are there any of these terms that you don’t know?

Binary trees

Root

Node

Arity of a node is the number of children that node has. In a binary tree, internal nodes have arity 1 or 2 and leaves have arity 0.

Leaf

Parent

Child

Subtree

Path

Depth (of node)

Height (of tree)

Level

Ancestor - Generalization of parent

Descendent - Generalization of child. It’s a property. (“a is a descendent of “b). (For all a s.t. a is a descendent of b …)

• A child of x is a descendent of x.
• If y is a descendent of x, then every child of y is also a descendent of x.

Forest - A collection of trees

Complete tree - All internal nodes have all children.

BSTs

Binary search tree (BST) is a tree (usually binary) of values (or key/value pairs) with the property that

• The keys of all the nodes in the left subtree are less than the key of the root.
• The keys of all the nodes in the right subtree are greater than the key of the root.
• The same property applies for all subtrees.

If we are using BSTs to implement dictionaries, each key appears only once.

If you have BSTs in which a value can appear multiple times, you can make one or both of the orderings not-strict.

Detour: Why BSTs?

Given that a balanced BST is still O(logn) for add and find, while a hash table is expected amortized O(1), why would I use BSTs rather than hash tables to implement dictionaries?

• They look pretty.
• Professors like to make students work with pointer structures.
• We want to think in terms of orderings.
  • For iteration or segmenting.
• Because there is no natural hash function.

Traversing trees

There are approximately eight standard orderings for traversing trees. What are they?

• Choice one: Recursively (depth-first) or across levels (breadth-first)
• Choice two: left-to-right or right-to-left
• Choice three for breadth-first: bottom-up or top down
• Choice three for depth-first: When do I look at the root value

How would you implement breadth-first, left-to-right, top-down traversal?

• We put nodes in a queue
• We repeatedly grab the front of the queue, process it, and add its children to the queue.

We can use a stack to do depth-first.

Building balanced BSTs

Suppose you have an array of key/value pairs and want to build a balanced binary tree from them.

Option one: After each insert, observe where it’s not balanced and reorganize.

Option two: Sort the key/value pairs. The middle one is the root. Recursively build the left and right halves.

Balancing BSTs

Think about this for next class: How do you keep your binary search trees balanced? (You don’t need them perfectly balanced; you just need to keep the depth O(logn))

• Approach one: Shift around nodes when it’s bad.
• Approach two: Reorganize