CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 1 Lecture 19 CS 1813 – Discrete Mathematics Trees and Inductive Definitions

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 2 What Is a Tree?  Tree  a diagram or graph that branches usually from a simple stem without forming loops or polygons Merriam-Webster  a type of data structure in which each element is attached to one or more elements directly beneath it — Webopedia  a node, together with a sequence of trees — inductive definition  Tree terminology  subtree — a node in a tree, together with its sequence of trees  root — the node that, with its subtrees, comprises the entire tree  interior node — a node with a nonempty sequence of subtrees  leaf — a node whose associated sequence of subtrees is empty  branch — a line connecting a node to its subtrees (in tree diagram)  binary tree — a tree with no nodes having more than 2 subtrees

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 3 What Are Trees Good For?  Computing applications that use trees  Databases, parsing, games, expert systems, word processors, operating systems, …  Search Trees — a common use of trees  Structures to make data quickly retrievable  Each node stores a key for retrieval and a package of data associated with the key  Keys are from a datatype that has an ordering  Subtrees are arranged to narrow the search quickly  Naïve search — just look sequentially through the pile Retrieval time proportional to number of items  Tree search — log(n) retrieval time, n = number items

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 4 Binary Search Tree – diagram form  Each node stores key and data  Left subtree contains all smaller keys  Right subtree contains all larger keys  Leafs just mark tree boundaries No data in a leaf 5120 PDA Cam 9605 Palm Pilot 4403 HotSync 4878 Palm Games 7268 Zip Drive 8444 Audio System 6876 Intellimouse 3663 Net Hub MB RAM K Modem 2088 LaserJet 1143 InkJet Key – ordered Data – anything To find an item start at root look left if smaller right if bigger 12 items: 4 steps, max

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 5 Binary Search Tree a formal representation data SearchTree key dat = Nub | Cel key dat (SearchTree key dat) (SearchTree key dat)  Type parameters — key, dat key — might be Int, for example (datatype with an ordering) dat — could be any type, typically a tuple  Example  s :: SearchTree Int (String, Float, [String] )  s is a SearchTree  key type – Int (maybe a catalog order number)  dat type – tuple storing a String (product description), a Float (price), and a sequence of strings (inventory records) Key goes hereNode data Left subtree (smaller keys) Right subtree (larger keys) Nub – leaf constructor Cel – constructor, non-empty trees

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 6 Retrieving Data from a Search Tree  Found or Not Found  Key sought may or may not be in search tree  What to do if it’s not there Deliver some sort of not-found indicator Use Maybe datatype for this data Maybe a = Just a | Nothing getItem :: Ord key => SearchTree key dat -> key -> Maybe (key, dat) getItem (Cel key dat smaller bigger) searchKey= if searchKey  key then (getItem smaller searchKey) else if searchKey  key then (getItem bigger searchKey) else (Just(key, dat)) getItem Nub searchKey = Nothing Not-Found IndicatorExample, item found Just (2088, “LaserJet”)

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 7 Tree Induction — another proof method  Definitions  Subtree s, t :: SearchTree key dat s  t  (s = t)  (  k,d,left,right. (t = Cel k d left right))  ((s  left)  (s  right))  Proper Subtree s, t :: SearchTree key dat s  t  (  k,d,left,right. (t = Cel k d left right))  ((s  left)  (s  right)) Equivalent Definition: s  t  s  t  s  t  Tree induction  P — predicate parameterized over SearchTrees P(t) is a proposition whenever t :: SearchTree key dat  Prove:  t. (  s  t. P(s))  P(t)  Conclude:  t. P(t)  Note: {s | s  Nub} = , so must prove P(Nub) directly (  s  Nub. P(s))  P(Nub) is equivalent to (True  P(Nub))

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 8 Properties of Trees  Definition “occurs in”  s :: SearchTree key dat, k :: key, d :: dat  k occurs in s — that is, k  s k  s  (  x,d,left,right. s  (Cel x d left right))  (k  x  k  left  k  right)  Definition “ordered” ordered s   (Cel k d left right)  s. (x  left  x  k)  (x  right  x  k)  Theorem: ordered trees have no duplicate keys P(s)  ((ordered s)  (x  s)  (s  (Cel k d left right)))  (x  k  (x  left  x  right))  (x  left  (x  k  x  right))  (x  right  (x  k  x  left)) Proof base case: s = Nub P(s) is True because the hypothesis of P(s) is False, since it cannot be the case that Nub  (Cel k d left right) and a thing constructed by Nub cannot be the same as a thing constructed by Cel inductive case: next slide

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 9 Ordered Trees Have No Duplicate Keys inductive case  Theorem: ordered trees have no duplicate keys P(s)  ((ordered s)  (x  s)  (s  (Cel k d left right)))  ((x  k  (x  left  x  right))  {1} (x  left  (x  k  x  right))  {2} (x  right  (x  k  x  left))) {3} Note: ordered s   (Cel k d left right)  s. (x  left  x  k)  (x  right  x  k) Proof inductive case: s = (Cel k d left right) 1. x  k   (x  k)   (x  k) {property of  and  }  x  left  x  right {ordered, contrapositive} 2. x  left  x  k   (x  k) {ordered, arithmetic}  x  k  x  right {arithmetic, ordered} 3. x  right  x  k   (x  k) {ordered, arithmetic}  x  k  x  left {arithmetic, ordered}  Conclusion:  s. P(s)

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 10 Proving That getItem Works  Theorem (getItem) ( (ordered s)  k  s )  getItem s k = Just (k, d)  Proof  P(s)  ((ordered s)  k  s)  getItem s k = Just (k, d)  Base case P(Nub)  ((ordered Nub)  k  Nub)  getItem Nub k = Just(k, d) The implication is true because its hypothesis is always false k  Nub would require that Nub = Cel x a left right A thing constructed by Nub cannot be the same as a thing constructed by Cel  Inductive case – next slide

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 11 getItem Proof — inductive case getItem :: Ord key => SearchTree key dat -> key -> Maybe (key, dat) getItem (Cel key dat smaller bigger) searchKey = if searchKey  key then (getItem smaller searchKey) else if searchKey  key then (getItem bigger searchKey) else (Just(key, dat)) getItem Nub searchKey = Nothing P(s)  ((ordered s)  k  s)   d. getItem s k = Just(k, d) Inductive Case: P(s), s  Nub  (ordered s)  k  s s = Cel x a left right s  Nub Case 1. x = k getItem s k = Just(k, a) x = k   (k  x)   (k  x) Case 2. k  x k  left s is ordered  d. getItem left k = Just(k, d) induction hyp: left  s  P(left) getItem s k = getItem left k searchKey  key branch of if-then getItem s k = Just(k, d) both sides = (getItem left k) Case 3. k  x — like Case 2, but use P(right)

CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page 12 End of Lecture