edits

[lambda.git] / manipulating_trees_with_monads.mdwn

diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn
index 1be499b..81dc451 100644 (file)
--- a/manipulating_trees_with_monads.mdwn
+++ b/manipulating_trees_with_monads.mdwn
@@ -1,10 +1,9 @@
 [[!toc]]
 
 [[!toc]]
 

-

 Manipulating trees with monads
 ------------------------------
 
 Manipulating trees with monads
 ------------------------------
 

-This thread develops an idea based on a detailed suggestion of Ken
+This topic develops an idea based on a detailed suggestion of Ken

 Shan's.  We'll build a series of functions that operate on trees,
 doing various things, including replacing leaves, counting nodes, and
 converting a tree to a list of leaves.  The end result will be an
 Shan's.  We'll build a series of functions that operate on trees,
 doing various things, including replacing leaves, counting nodes, and
 converting a tree to a list of leaves.  The end result will be an


@@ -12,295 +11,284 @@ application for continuations.
 
 From an engineering standpoint, we'll build a tree transformer that
 deals in monads.  We can modify the behavior of the system by swapping
 
 From an engineering standpoint, we'll build a tree transformer that
 deals in monads.  We can modify the behavior of the system by swapping

-one monad for another.  (We've already seen how adding a monad can add
+one monad for another.  We've already seen how adding a monad can add

 a layer of funtionality without disturbing the underlying system, for
 instance, in the way that the reader monad allowed us to add a layer
 of intensionality to an extensional grammar, but we have not yet seen
 a layer of funtionality without disturbing the underlying system, for
 instance, in the way that the reader monad allowed us to add a layer
 of intensionality to an extensional grammar, but we have not yet seen

-the utility of replacing one monad with other.)
+the utility of replacing one monad with other.

 
 

-First, we'll be needing a lot of trees during the remainder of the
-course.  Here's a type constructor for binary trees:
+First, we'll be needing a lot of trees for the remainder of the
+course.  Here again is a type constructor for leaf-labeled, binary trees:

 
     type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
 
 
     type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
 

-These are trees in which the internal nodes do not have labels.  [How
-would you adjust the type constructor to allow for labels on the
+[How would you adjust the type constructor to allow for labels on the

 internal nodes?]
 
 We'll be using trees where the nodes are integers, e.g.,
 
 
 internal nodes?]
 
 We'll be using trees where the nodes are integers, e.g.,
 
 

-<pre>
-let t1 = Node ((Node ((Leaf 2), (Leaf 3))),
-               (Node ((Leaf 5),(Node ((Leaf 7),
-                                      (Leaf 11))))))
-
-    .
- ___|___
- |     |
- .     .
-_|__  _|__
-|  |  |  |
-2  3  5  .
-        _|__
-        |  |
-        7  11
-</pre>
+       let t1 = Node (Node (Leaf 2, Leaf 3),
+                      Node (Leaf 5, Node (Leaf 7,
+                                             Leaf 11)))
+           .
+        ___|___
+        |     |
+        .     .
+       _|_   _|__
+       |  |  |  |
+       2  3  5  .
+               _|__
+               |  |
+               7  11

 
 Our first task will be to replace each leaf with its double:
 
 
 Our first task will be to replace each leaf with its double:
 

-<pre>
-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));;
-</pre>
-`treemap` takes a function that transforms old leaves into new leaves, 
+       let rec tree_map (leaf_modifier : 'a -> 'b) (t : 'a tree) : 'b tree =
+         match t with
+           | Leaf i -> Leaf (leaf_modifier i)
+           | Node (l, r) -> Node (tree_map leaf_modifier l,
+                                  tree_map leaf_modifier r);;
+
+`tree_map` takes a function that transforms old leaves into new leaves,

 and maps that function over all the leaves in the tree, leaving the
 structure of the tree unchanged.  For instance:
 
 and maps that function over all the leaves in the tree, leaving the
 structure of the tree unchanged.  For instance:
 

-<pre>
-let double i = i + i;;
-treemap double t1;;
-- : int tree =
-Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
-
-    .
- ___|____
- |      |
- .      .
-_|__  __|__
-|  |  |   |
-4  6  10  .
-        __|___
-        |    |
-        14   22
-</pre>
-
-We could have built the doubling operation right into the `treemap`
-code.  However, because what to do to each leaf is a parameter, we can
+       let double i = i + i;;
+       tree_map double t1;;
+       - : int tree =
+       Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
+       
+           .
+        ___|____
+        |      |
+        .      .
+       _|__  __|__
+       |  |  |   |
+       4  6  10  .
+               __|___
+               |    |
+               14   22
+
+We could have built the doubling operation right into the `tree_map`
+code.  However, because we've left what to do to each leaf as a parameter, we can

 decide to do something else to the leaves without needing to rewrite
 decide to do something else to the leaves without needing to rewrite

-`treemap`.  For instance, we can easily square each leaf instead by
+`tree_map`.  For instance, we can easily square each leaf instead by

 supplying the appropriate `int -> int` operation in place of `double`:
 
 supplying the appropriate `int -> int` operation in place of `double`:
 

-<pre>
-let square x = x * x;;
-treemap square t1;;
-- : int tree =ppp
-Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
-</pre>
+       let square i = i * i;;
+       tree_map square t1;;
+       - : int tree =ppp
+       Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))

 
 

-Note that what `treemap` does is take some global, contextual
+Note that what `tree_map` does is take some global, contextual

 information---what to do to each leaf---and supplies that information
 information---what to do to each leaf---and supplies that information

-to each subpart of the computation.  In other words, `treemap` has the
-behavior of a reader monad.  Let's make that explicit.  
+to each subpart of the computation.  In other words, `tree_map` has the
+behavior of a reader monad.  Let's make that explicit.

 
 

-In general, we're on a journey of making our treemap function more and
-more flexible.  So the next step---combining the tree transducer with
-a reader monad---is to have the treemap function return a (monadized)
-tree that is ready to accept any `int->int` function and produce the
+In general, we're on a journey of making our `tree_map` function more and
+more flexible.  So the next step---combining the tree transformer with
+a reader monad---is to have the `tree_map` function return a (monadized)
+tree that is ready to accept any `int -> int` function and produce the

 updated tree.
 
 updated tree.
 

-\tree (. (. (f2) (f3))(. (f5) (.(f7)(f11))))
-<pre>
-\f    .
-  ____|____
-  |       |
-  .       .
-__|__   __|__
-|   |   |   |
-f2  f3  f5  .
-          __|___
-          |    |
-          f7  f11
-</pre>
+\tree (. (. (f 2) (f 3)) (. (f 5) (. (f 7) (f 11))))
+
+       \f      .
+          _____|____
+          |        |
+          .        .
+        __|___   __|___
+        |    |   |    |
+       f 2  f 3  f 5  .
+                    __|___
+                    |    |
+                   f 7  f 11

 
 That is, we want to transform the ordinary tree `t1` (of type `int
 
 That is, we want to transform the ordinary tree `t1` (of type `int

-tree`) into a reader object of type `(int->int)-> int tree`: something 
-that, when you apply it to an `int->int` function returns an `int
-tree` in which each leaf `x` has been replaced with `(f x)`.
+tree`) into a reader object of type `(int -> int) -> int tree`: something
+that, when you apply it to an `int -> int` function `f` returns an `int
+tree` in which each leaf `i` has been replaced with `f i`.

 
 With previous readers, we always knew which kind of environment to
 expect: either an assignment function (the original calculator
 simulation), a world (the intensionality monad), an integer (the
 
 With previous readers, we always knew which kind of environment to
 expect: either an assignment function (the original calculator
 simulation), a world (the intensionality monad), an integer (the

-Jacobson-inspired link monad), etc.  In this situation, it will be
-enough for now to expect that our reader will expect a function of
-type `int->int`.
-
-<pre>
-type 'a reader = (int->int) -> 'a;;  (* mnemonic: e for environment *)
-let reader_unit (x:'a): 'a reader = fun _ -> x;;
-let reader_bind (u: 'a reader) (f:'a -> 'c reader):'c reader = fun e -> f (u e) e;;
-</pre>
-
-It's easy to figure out how to turn an `int` into an `int reader`:
-
-<pre>
-let int2int_reader (x:'a): 'b reader = fun (op:'a -> 'b) -> op x;;
-int2int_reader 2 (fun i -> i + i);;
-- : int = 4
-</pre>
-
-But what do we do when the integers are scattered over the leaves of a
-tree?  A binary tree is not the kind of thing that we can apply a
-function of type `int->int` to.
-
-<pre>
-let rec treemonadizer (f:'a -> 'b reader) (t:'a tree):('b tree) reader =
-  match t with Leaf x -> reader_bind (f x) (fun x' -> reader_unit (Leaf x'))
-             | Node (l, r) -> reader_bind (treemonadizer f l) (fun x ->
-                                reader_bind (treemonadizer f r) (fun y ->
-                                  reader_unit (Node (x, y))));;
-</pre>
+Jacobson-inspired link monad), etc.  In the present case, it will be
+enough to expect that our "environment" will be some function of type
+`int -> int`.
+
+       type 'a reader = (int -> int) -> 'a;;  (* mnemonic: e for environment *)
+       let reader_unit (a : 'a) : 'a reader = fun _ -> a;;
+       let reader_bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader = fun e -> f (u e) e;;
+
+It would be a simple matter to turn an *integer* into an `int reader`:
+
+       let int_readerize : int -> int reader = fun (a : int) -> fun (modifier : int -> int) -> modifier a;;
+       int_readerize 2 (fun i -> i + i);;
+       - : int = 4
+
+But how do we do the analagous transformation when our `int`s are scattered over the leaves of a tree? How do we turn an `int tree` into a reader?
+A tree is not the kind of thing that we can apply a
+function of type `int -> int` to.
+
+But we can do this:
+
+       let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader =
+           match t with
+           | Leaf i -> reader_bind (f i) (fun i' -> reader_unit (Leaf i'))
+           | Node (l, r) -> reader_bind (tree_monadize f l) (fun x ->
+                              reader_bind (tree_monadize f r) (fun y ->
+                                reader_unit (Node (x, y))));;

 
 This function says: give me a function `f` that knows how to turn
 
 This function says: give me a function `f` that knows how to turn

-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
+something of type `'a` into an `'b reader`, and I'll show you how to
+turn an `'a tree` into an `'b tree reader`.  In more fanciful terms,
+the `tree_monadize` function builds plumbing that connects all of the

 leaves of a tree into one connected monadic network; it threads the
 leaves of a tree into one connected monadic network; it threads the

-monad through the leaves.
+`'b reader` monad through the original tree's leaves.

 
 

-<pre>
-# treemonadizer int2int_reader t1 (fun i -> i + i);;
-- : int tree =
-Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
-</pre>
+       # tree_monadize int_readerize t1 double;;
+       - : int tree =
+       Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))

 
 Here, our environment is the doubling function (`fun i -> i + i`).  If
 
 Here, our environment is the doubling function (`fun i -> i + i`).  If

-we apply the very same `int tree reader` (namely, `treemonadizer
-int2int_reader t1`) to a different `int->int` function---say, the
+we apply the very same `int tree reader` (namely, `tree_monadize
+int_readerize t1`) to a different `int -> int` function---say, the

 squaring function, `fun i -> i * i`---we get an entirely different
 result:
 
 squaring function, `fun i -> i * i`---we get an entirely different
 result:
 

-<pre>
-# treemonadizer int2int_reader t1 (fun i -> i * i);;
-- : int tree =
-Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
-</pre>
+       # tree_monadize int_readerize t1 square;;
+       - : int tree =
+       Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))

 
 

-Now that we have a tree transducer 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.
 parameter, we can see what it would take to swap in a different monad.

+
+<!-- FIXME -->
+

 For instance, we can use a state monad to count the number of nodes in
 the tree.
 
 For instance, we can use a state monad to count the number of nodes in
 the tree.
 

-<pre>
-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);;
-</pre>
+       type 'a state = int -> 'a * int;;
+       let state_unit a = fun s -> (a, s);;
+       let state_bind_and_count u f = fun s -> let (a, s') = u s in f a (s' + 1);;

 
 

-Gratifyingly, we can use the `treemonadizer` function without any
+Gratifyingly, we can use the `tree_monadize` function without any

 modification whatsoever, except for replacing the (parametric) type
 modification whatsoever, except for replacing the (parametric) type

-`reader` with `state`:
+`'b reader` with `'b state`, and substituting in the appropriate unit and bind:

 
 

-<pre>
-let rec treemonadizer (f:'a -> 'b state) (t:'a tree):('b tree) state =
-  match t with Leaf x -> state_bind (f x) (fun x' -> state_unit (Leaf x'))
-             | Node (l, r) -> state_bind (treemonadizer f l) (fun x ->
-                                state_bind (treemonadizer f r) (fun y ->
-                                  state_unit (Node (x, y))));;
-</pre>
+       let rec tree_monadize (f : 'a -> 'b state) (t : 'a tree) : 'b tree state =
+           match t with
+           | Leaf i -> state_bind_and_count (f i) (fun i' -> state_unit (Leaf i'))
+           | Node (l, r) -> state_bind_and_count (tree_monadize f l) (fun x ->
+                              state_bind_and_count (tree_monadize f r) (fun y ->
+                                state_unit (Node (x, y))));;

 
 Then we can count the number of nodes in the tree:
 
 
 Then we can count the number of nodes in the tree:
 

-<pre>
-# treemonadizer state_unit t1 0;;
-- : int tree * int =
-(Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13)
-
-    .
- ___|___
- |     |
- .     .
-_|__  _|__
-|  |  |  |
-2  3  5  .
-        _|__
-        |  |
-        7  11
-</pre>
+       # tree_monadize state_unit t1 0;;
+       - : int tree * int =
+       (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13)
+       
+           .
+        ___|___
+        |     |
+        .     .
+       _|__  _|__
+       |  |  |  |
+       2  3  5  .
+               _|__
+               |  |
+               7  11

 
 Notice that we've counted each internal node twice---it's a good
 exercise to adjust the code to count each node once.
 
 
 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
 One more revealing example before getting down to business: replacing

-`state` everywhere in `treemonadizer` with `list` gives us
+`state` everywhere in `tree_monadize` with `list` gives us

 
 

-<pre>
-# treemonadizer (fun x -> [ [x; square x] ]) t1;;
-- : int list tree list =
-[Node
-  (Node (Leaf [2; 4], Leaf [3; 9]),
-   Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))]
-</pre>
+       # tree_monadize (fun i -> [ [i; square i] ]) t1;;
+       - : int list tree list =
+       [Node
+         (Node (Leaf [2; 4], Leaf [3; 9]),
+          Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))]

 
 Unlike the previous cases, instead of turning a tree into a function
 from some input to a result, this transformer replaces each `int` with
 a list of `int`'s.
 
 
 Unlike the previous cases, instead of turning a tree into a function
 from some input to a result, this transformer replaces each `int` with
 a list of `int`'s.
 

-Now for the main point.  What if we wanted to convert a tree to a list
-of leaves?  
+<!--
+We don't make it clear why the fun has to be int -> int list list, instead of int -> int list
+-->

 
 

-<pre>
-type ('a, 'r) continuation = ('a -> 'r) -> 'r;;
-let continuation_unit x c = c x;;
-let continuation_bind u f c = u (fun a -> f a c);;

 
 

-let rec treemonadizer (f:'a -> ('b, 'r) continuation) (t:'a tree):(('b tree), 'r) continuation =
-  match t with Leaf x -> continuation_bind (f x) (fun x' -> continuation_unit (Leaf x'))
-             | Node (l, r) -> continuation_bind (treemonadizer f l) (fun x ->
-                                continuation_bind (treemonadizer f r) (fun y ->
-                                  continuation_unit (Node (x, y))));;
-</pre>
+Now for the main point.  What if we wanted to convert a tree to a list
+of leaves?
+
+       type ('a, 'r) continuation = ('a -> 'r) -> 'r;;
+       let continuation_unit a = fun k -> k a;;
+       let continuation_bind u f = fun k -> u (fun a -> f a k);;
+       
+       let rec tree_monadize (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation =
+           match t with
+           | Leaf i -> continuation_bind (f i) (fun i' -> continuation_unit (Leaf i'))
+           | Node (l, r) -> continuation_bind (tree_monadize f l) (fun x ->
+                              continuation_bind (tree_monadize f r) (fun y ->
+                                continuation_unit (Node (x, y))));;

 
 We use the continuation monad described above, and insert the
 
 We use the continuation monad described above, and insert the

-`continuation` type in the appropriate place in the `treemonadizer` code.
+`continuation` type in the appropriate place in the `tree_monadize` code.

 We then compute:
 
 We then compute:
 

-<pre>
-# treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);;
-- : int list = [2; 3; 5; 7; 11]
-</pre>
+       # tree_monadize (fun a k -> a :: (k a)) t1 (fun t -> []);;
+       - : int list = [2; 3; 5; 7; 11]
+
+<!-- FIXME: what if we had fun t -> [-t]? why `t`? -->

 
 We have found a way of collapsing a tree into a list of its leaves.
 
 The continuation monad is amazingly flexible; we can use it to
 simulate some of the computations performed above.  To see how, first
 
 We have found a way of collapsing a tree into a list of its leaves.
 
 The continuation monad is amazingly flexible; we can use it to
 simulate some of the computations performed above.  To see how, first

-note that an interestingly uninteresting thing happens if we use the
-continuation unit as our first argument to `treemonadizer`, and then
+note that an interestingly uninteresting thing happens if we use
+`continuation_unit` as our first argument to `tree_monadize`, and then

 apply the result to the identity function:
 
 apply the result to the identity function:
 

-<pre>
-# treemonadizer continuation_unit t1 (fun x -> x);;
-- : int tree =
-Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
-</pre>
+       # tree_monadize continuation_unit t1 (fun i -> i);;
+       - : int tree =
+       Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))

 
 That is, nothing happens.  But we can begin to substitute more
 
 That is, nothing happens.  But we can begin to substitute more

-interesting functions for the first argument of `treemonadizer`:
-
-<pre>
-(* Simulating the tree reader: distributing a operation over the leaves *)
-# treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);;
-- : int tree =
-Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
-
-(* Simulating the int list tree list *)
-# treemonadizer (fun a c -> c [a; square a]) t1 (fun x -> x);;
-- : int list tree =
-Node
- (Node (Leaf [2; 4], Leaf [3; 9]),
-  Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))
-
-(* Counting leaves *)
-# treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);;
-- : int = 5
-</pre>
+interesting functions for the first argument of `tree_monadize`:
+
+       (* Simulating the tree reader: distributing a operation over the leaves *)
+       # tree_monadize (fun a k -> k (square a)) t1 (fun i -> i);;
+       - : int tree =
+       Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
+
+       (* Simulating the int list tree list *)
+       # tree_monadize (fun a k -> k [a; square a]) t1 (fun i -> i);;
+       - : int list tree =
+       Node
+        (Node (Leaf [2; 4], Leaf [3; 9]),
+         Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))
+
+       (* Counting leaves *)
+       # tree_monadize (fun a k -> 1 + k a) t1 (fun i -> 0);;
+       - : int = 5

 
 We could simulate the tree state example too, but it would require
 
 We could simulate the tree state example too, but it would require

-generalizing the type of the continuation monad to 
+generalizing the type of the continuation monad to

 
 

-    type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;;
+       type ('a, 'b, 'c) continuation = ('a -> 'b) -> 'c;;

 
 The binary tree monad
 ---------------------
 
 The binary tree monad
 ---------------------


@@ -308,79 +296,74 @@ The binary tree monad
 Of course, by now you may have realized that we have discovered a new
 monad, the binary tree monad:
 
 Of course, by now you may have realized that we have discovered a new
 monad, the binary tree monad:
 

-<pre>
-type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
-let tree_unit (x:'a) = Leaf x;;
-let rec tree_bind (u:'a tree) (f:'a -> 'b tree):'b tree = 
-  match u with Leaf x -> f x 
-             | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));;
-</pre>
+       type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
+       let tree_unit (a: 'a) = Leaf a;;
+       let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree =
+           match u with
+           | Leaf a -> f a
+           | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));;

 
 For once, let's check the Monad laws.  The left identity law is easy:
 
 
 For once, let's check the Monad laws.  The left identity law is easy:
 

-    Left identity: bind (unit a) f = bind (Leaf a) f = fa
+    Left identity: bind (unit a) f = bind (Leaf a) f = f a

 
 To check the other two laws, we need to make the following
 observation: it is easy to prove based on `tree_bind` by a simple
 induction on the structure of the first argument that the tree
 resulting from `bind u f` is a tree with the same strucure as `u`,
 
 To check the other two laws, we need to make the following
 observation: it is easy to prove based on `tree_bind` by a simple
 induction on the structure of the first argument that the tree
 resulting from `bind u f` is a tree with the same strucure as `u`,

-except that each leaf `a` has been replaced with `fa`:
-
-\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5)))
-<pre>
-                .                         .       
-              __|__                     __|__   
-              |   |                     |   |   
-              a1  .                    fa1  .   
-                 _|__                     __|__ 
-                 |  |                     |   | 
-                 .  a5                    .  fa5
-   bind         _|__       f   =        __|__   
-                |  |                    |   |   
-                .  a4                   .  fa4  
-              __|__                   __|___   
-              |   |                   |    |   
-              a2  a3                 fa2  fa3         
-</pre>   
+except that each leaf `a` has been replaced with `f a`:
+
+\tree (. (f a1) (. (. (. (f a2) (f a3)) (f a4)) (f a5)))
+
+                       .                         .
+                     __|__                     __|__
+                     |   |                     |   |
+                     a1  .                   f a1  .
+                        _|__                     __|__
+                        |  |                     |   |
+                        .  a5                    .  f a5
+          bind         _|__       f   =        __|__
+                       |  |                    |   |
+                       .  a4                   .  f a4
+                     __|__                   __|___
+                     |   |                   |    |
+                     a2  a3                f a2  f a3

 
 Given this equivalence, the right identity law
 
 
 Given this equivalence, the right identity law
 

-    Right identity: bind u unit = u
+       Right identity: bind u unit = u

 
 falls out once we realize that
 
 
 falls out once we realize that
 

-    bind (Leaf a) unit = unit a = Leaf a
+       bind (Leaf a) unit = unit a = Leaf a

 
 As for the associative law,
 
 
 As for the associative law,
 

-    Associativity: bind (bind u f) g = bind u (\a. bind (fa) g)
+       Associativity: bind (bind u f) g = bind u (\a. bind (f a) g)

 
 we'll give an example that will show how an inductive proof would
 proceed.  Let `f a = Node (Leaf a, Leaf a)`.  Then
 
 
 we'll give an example that will show how an inductive proof would
 proceed.  Let `f a = Node (Leaf a, Leaf a)`.  Then
 

-\tree (. (. (. (. (a1)(a2)))))
-\tree (. (. (. (. (a1) (a1)) (. (a1) (a1)))  ))
-<pre>
-                                           .
-                                       ____|____
-          .               .            |       |
-bind    __|__   f  =    __|_    =      .       .
-        |   |           |   |        __|__   __|__
-        a1  a2         fa1 fa2       |   |   |   |
-                                     a1  a1  a1  a1  
-</pre>
+\tree (. (. (. (. (a1) (a2)))))
+\tree (. (. (. (. (a1) (a1)) (. (a1) (a1)))))
+
+                                                  .
+                                              ____|____
+                 .               .            |       |
+       bind    __|__   f  =    __|_    =      .       .
+               |   |           |   |        __|__   __|__
+               a1  a2        f a1 f a2      |   |   |   |
+                                            a1  a1  a1  a1

 
 Now when we bind this tree to `g`, we get
 
 
 Now when we bind this tree to `g`, we get
 

-<pre>
-           .
-       ____|____
-       |       |
-       .       .
-     __|__   __|__
-     |   |   |   |
-    ga1 ga1 ga1 ga1  
-</pre>
+                   .
+              _____|______
+              |          |
+              .          .
+            __|__      __|__
+            |   |      |   |
+          g a1 g a1  g a1 g a1

 
 At this point, it should be easy to convince yourself that
 using the recipe on the right hand side of the associative law will
 
 At this point, it should be easy to convince yourself that
 using the recipe on the right hand side of the associative law will


@@ -391,6 +374,5 @@ So binary trees are a monad.
 Haskell combines this monad with the Option monad to provide a monad
 called a
 [SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree)
 Haskell combines this monad with the Option monad to provide a monad
 called a
 [SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree)

-that is intended to 
-represent non-deterministic computations as a tree.
+that is intended to represent non-deterministic computations as a tree.