Functional%20Programming%20Lecture%207%20-%20Trees

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Functional ProgrammingLecture 7 - Trees
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Binary Trees of Numbers
42
13
19
12

• data NTree NilT
• Node Int NTree NTree
• Node 42
• (Node 13 NilT NilT)
• (Node 19 NilT (Node 12 NilT NilT))

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Operations on Binary Trees of Numbers

• Most operations can be defined using primitive
  recursion and pattern matching.
• depth Ntree -gt Int
• -- calculate the depth of a tree
• depth NilT 0
• depth (Node n t1 t2) 1 max (depth t1) (depth
  t2)
• sumtree Ntree -gt Int
• -- sum all the nodes of a tree
• sumtree NilT 0
• sumtree (Node n t1 t2)
• n (sumtree t1) (sumtree t2)
• Evaluate on tree 42
• 13 19
• 
  12

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Operations on Binary Trees of Numbers

• occurs Int -gt Ntree -gt Int
• -- occurrences of a number in a tree
• occurs x NilT 0
• occurs x (Node n t1 t2)
• (x n) 1 (occurs x t1) (occurs x
  t2)
• otherwise (occurs x t1) (occurs x
  t2)
• occurs on lists?
• occurs Int -gt Int -gt Int
• occurs x 0
• occurs x (yys)
• (x y) 1(occurs x ys)
• otherwise occurs x ys

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• left Ntree -gt Ntree
• -- left subtree
• left NilT NilT
• left (Node n t1 t2) t1
• right Ntree -gt Ntree
• -- right subtree
• right NilT NilT
• right (Node n t1 t2) t2

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Operations on Binary Trees of Numbers

• Conjoin two trees
• 
  6
• 2 3 gt
  2 3
• 4 6 1 5 4
  6 1 5
• conjoin Int -gt Ntree -gt Ntree -gt Ntree
• -- conjoin two trees
• conjoin x t1 t2 Node x t1 t2

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Operations on Binary Trees of Numbers

• maxt Ntree -gt Int
• -- find max value in a tree
• maxt NilT ??
• maxt(Node n NilT NilT) n
• maxt(Node n NilT t2) max n (maxt t2)
• maxt(Node n t1 NilT) max n (maxt t1)
• maxt (Node n t1 t2) max3 n (maxt t1) (maxt t2)
• or
• maxt t maxlist preorder t
• where maxlist x x
• maxlist (xxs) max x (maxlist xs)
  

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Operations on Binary Trees of Numbers

• or
• maxt Ntree -gt Maybe Int
• maxt NilT Nothing
• maxt(Node n NilT NilT) Just n
• maxt(Node n NilT t2) max n y
• maxt(Node n t1 NilT) max n x
• maxt (Node n t1 t2) Just max3 n x y
• where Just x maxt t1
• Just y maxt t2

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Traversing Binary Trees of Numbers

• preorder Ntree -gt Int
• preorder NilT
• preorder (Node n t1 t2) n(preorder t1
  preorder t2)
• Example 42
• 3 8
• 16 5
• preorder (Node 42 (Node 3 Nil Nil) (Node 8 (Node
  16 Nil Nil) (Node 5 Nil Nil)))

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Traversing Binary Trees of Numbers

• inorder Ntree -gt Int
• inorder NilT
• inorder (Node n t1 t2)
• inorder t1 n inorder t2
• Example 42
• 3 8
• 16 5
• inorder (Node 42 (Node 3 Nil Nil) (Node 8 (Node
  16 Nil Nil) (Node 5 Nil Nil)))

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Traversing Binary Trees of Numbers

• postorder Ntree -gt Int
• postorder NilT
• postorder (Node n t1 t2)
• postorder t1 postorder t2 n
• Example 42
• 3 8
• 16 5

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Sorting Binary Trees of Numbers

• sorttree Ntree -gt Int
• sorttree t sort (preorder t)
• where sort is your favourite sorting algorithm.

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Insertion sort
sort xs inssort(xs,) a common programming
paradigm inssort (Int,Int) -gt Int --
insertion sort -- first component is list to be
sorted -- second component is sorted list inssort
(,ys) ys inssort (xxs, ys) inssort(xs, ins
x ys) ins Int -gtInt -gt Int -- insert
integer at correct position in sorted list ins x
x ins x (yys) if (xlty) then (xyys)
else y (ins x ys) sort( 3,2,1
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Map

• maptree (Int -gt Int) -gt Ntree -gt Ntree
• maptree f NilT NilT
• maptree f (Node n t1 t2)
• Node (f n) (maptree f t1) (maptree f t2)
• Example t 42
• 3 8
• 16 5
• maptree times2 t

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Polymorphic Binary Trees

• We can define trees with an arbitrary type at the
  nodes
• data Tree a Nil
• Node a (Tree a) (Tree
  a)
• deriving Show
• So define
• maptree (a -gt b) -gt (Tree a) -gt (Tree b)
• maptree Nil Nil
• maptree f (Node n t1 t2)
• Node f n (maptree f t1) (maptree f t2)
• preorder Tree a -gt a
• Etc.
• What is type of maptree times2?