From: Jim Pryor <profjim@jimpryor.net>
Date: Wed, 1 Dec 2010 08:23:12 +0000 (-0500)
Subject: manip trees: tweaks
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=26319cf2ffc188af7fc324143881d45fd7c322c8

manip trees: tweaks

Signed-off-by: Jim Pryor <profjim@jimpryor.net>
---

diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn
index 20039c28..de8fc5fa 100644
--- a/manipulating_trees_with_monads.mdwn
+++ b/manipulating_trees_with_monads.mdwn
@@ -28,9 +28,9 @@ internal nodes?]
 We'll be using trees where the nodes are integers, e.g.,
 
 
-	let t1 = Node ((Node ((Leaf 2), (Leaf 3))),
-	               (Node ((Leaf 5),(Node ((Leaf 7),
-	                                      (Leaf 11))))))
+	let t1 = Node (Node (Leaf 2, Leaf 3),
+	               Node (Leaf 5, Node (Leaf 7,
+	                                      Leaf 11)))
 	    .
 	 ___|___
 	 |     |
@@ -47,8 +47,8 @@ Our first task will be to replace each leaf with its double:
 	let rec treemap (newleaf : 'a -> 'b) (t : 'a tree) : 'b tree =
 	  match t with
 	    | Leaf x -> Leaf (newleaf x)
-	    | Node (l, r) -> Node ((treemap newleaf l),
-	                           (treemap newleaf r));;
+	    | Node (l, r) -> Node (treemap newleaf l,
+	                           treemap newleaf r);;
 
 `treemap` takes a function that transforms old leaves into new leaves,
 and maps that function over all the leaves in the tree, leaving the
@@ -122,7 +122,7 @@ type `int -> int`.
 
 It's easy to figure out how to turn an `int` into an `int reader`:
 
-	let int2int_reader (x : 'a) : 'b reader = fun (op : 'a -> 'b) -> op x;;
+	let int2int_reader : 'a -> 'b reader = fun (a : 'a) -> fun (op : 'a -> 'b) -> op a;;
 	int2int_reader 2 (fun i -> i + i);;
 	- : int = 4
 
@@ -142,7 +142,7 @@ something of type `'a` into an `'b reader`, and I'll show you how to
 turn an `'a tree` into an `'a tree reader`.  In more fanciful terms,
 the `treemonadizer` function builds plumbing that connects all of the
 leaves of a tree into one connected monadic network; it threads the
-monad through the leaves.
+`'b reader` monad through the leaves.
 
 	# treemonadizer int2int_reader t1 (fun i -> i + i);;
 	- : int tree =
@@ -158,18 +158,18 @@ result:
 	- : int tree =
 	Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
 
-Now that we have a tree transformer that accepts a monad as a
+Now that we have a tree transformer that accepts a reader monad as a
 parameter, we can see what it would take to swap in a different monad.
 For instance, we can use a state monad to count the number of nodes in
 the tree.
 
 	type 'a state = int -> 'a * int;;
-	let state_unit x i = (x, i+.5);;
-	let state_bind u f i = let (a, i') = u i in f a (i'+.5);;
+	let state_unit a = fun i -> (a, i);;
+	let state_bind u f = fun i -> let (a, i') = u i in f a (i' + 1);;
 
 Gratifyingly, we can use the `treemonadizer` function without any
 modification whatsoever, except for replacing the (parametric) type
-`reader` with `state`:
+`'b reader` with `'b state`, and substituting in the appropriate unit and bind:
 
 	let rec treemonadizer (f : 'a -> 'b state) (t : 'a tree) : 'b tree state =
 	    match t with
@@ -198,6 +198,15 @@ Then we can count the number of nodes in the tree:
 Notice that we've counted each internal node twice---it's a good
 exercise to adjust the code to count each node once.
 
+<!--
+A tree with n leaves has 2n - 1 nodes.
+This function will currently return n*1 + (n-1)*2 = 3n - 2.
+To convert b = 3n - 2 into 2n - 1, we can use: let n = (b + 2)/3 in 2*n -1
+
+But I assume Chris means here, adjust the code so that no corrections of this sort have to be applied.
+-->
+
+
 One more revealing example before getting down to business: replacing
 `state` everywhere in `treemonadizer` with `list` gives us