XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week11.mdwn;h=14d5eed2745a5ab8cb140fc76c09f477d93d47c6;hp=79543c8ac68c0be818bb1b076ad8670eb28adfcc;hb=8fbf14566363197bb08307a6bf4daece2f27602e;hpb=671e314f91abff10acfa31eab2211bb7eeb17881
diff git a/week11.mdwn b/week11.mdwn
index 79543c8a..14d5eed2 100644
 a/week11.mdwn
+++ b/week11.mdwn
@@ 111,7 +111,7 @@ The leftmost leaf and the rightmost leaf have the same label; but they are diffe
/ \
3 4
Here I haven't drawn what the labels are. The leftmost leaf, the node tagged "3" in this diagram, doesn't have the label `3`. It has the label 1, as we said before. I just haven't put that into the diagram. The node tagged "2" doesn't have the label `2`. It doesn't have any label. The tree in this example only has labels on its leafs, not on any of its inner nodes.
+Here I haven't drawn what the labels are. The leftmost leaf, the node tagged "3" in this diagram, doesn't have the label `3`. It has the label 1, as we said before. I just haven't put that into the diagram. The node tagged "2" doesn't have the label `2`. It doesn't have any label. The tree in this example only has information labeling its leafs, not any of its inner nodes. The identity of its inner nodes is exhausted by their position in the tree.
That is a second thing to note. In what follows, we'll only be working with *leaflabeled* trees. In some uses of trees, one also wants labels on inner nodes. But we won't be discussing any such trees now. Our trees only have labels on their leafs. The diagrams below will tag all of the nodes, as in the second diagram above, and won't display what the leafs' labels are.
@@ 134,21 +134,21 @@ This is a leaflabeled tree whose labels aren't displayed. The `9200` and so on
Suppose we want to represent that we're *at* the node marked `50`. We might use the following metalanguage notation to specify this:
 {parent = ...; siblings = [node 20; *; node 80]}, * filled by node 50
+ {parent = ...; siblings = [subtree 20; *; subtree 80]}, * filled by subtree 50
This is modeled on the notation suggested above for list zippers. Here `node 20` refers not to a `int` label associated with that node, but rather to the whole subtree rooted at that node:
+This is modeled on the notation suggested above for list zippers. Here `subtree 20` refers to the whole subtree rooted at node `20`:
20
/  \
1 2 3
Similarly for `node 50` and `node 80`. We haven't said yet what goes in the `parent = ...` slot. Well, the parent of a subtree targetted on `node 50` should intuitively be a tree targetted on `node 500`:
+Similarly for `subtree 50` and `subtree 80`. We haven't said yet what goes in the `parent = ...` slot. Well, the parent of a subtree targetted on `node 50` should intuitively be a tree targetted on `node 500`:
 {parent = ...; siblings = [*; node 920; node 950]}, * filled by node 500
+ {parent = ...; siblings = [*; subtree 920; subtree 950]}, * filled by subtree 500
And the parent of that targetted subtree should intuitively be a tree targetted on `node 9200`:
 {parent = None; siblings = [*]}, * filled by node 9200
+ {parent = None; siblings = [*]}, * filled by tree 9200
This tree has no parents because it's the root of the base tree. Fully spelled out, then, our tree targetted on `node 50` would be:
@@ 157,26 +157,13 @@ This tree has no parents because it's the root of the base tree. Fully spelled o
parent = {
parent = None;
siblings = [*]
 }, * filled by node 9200;
 siblings = [*; node 920; node 950]
 }, * filled by node 500;
 siblings = [node 20; *; node 80]
 }, * filled by node 50
+ }, * filled by tree 9200;
+ siblings = [*; subtree 920; subtree 950]
+ }, * filled by subtree 500;
+ siblings = [subtree 20; *; subtree 80]
+ }, * filled by subtree 50
In fact, there's some redundancy in this structure, at the points where we have `* filled by node 9200` and `* filled by node 500`. Most of `node 9200`with the exception of any label attached to node `9200` itselfis determined by the rest of this structure; and so too with `node 500`. So we could really work with:

 {
 parent = {
 parent = {
 parent = None;
 siblings = [*]
 }, label for * position (at node 9200);
 siblings = [*; node 920; node 950]
 }, label for * position (at node 500);
 siblings = [node 20; *; node 80]
 }, * filled by node 50

Or, if we only had labels on the leafs of our tree:
+In fact, there's some redundancy in this structure, at the points where we have `* filled by tree 9200` and `* filled by subtree 500`. Since node 9200 doesn't have any label attached to it, the subtree rooted in it is determined by the rest of this structure; and so too with `subtree 500`. So we could really work with:
{
parent = {
@@ 184,47 +171,46 @@ Or, if we only had labels on the leafs of our tree:
parent = None;
siblings = [*]
},
 siblings = [*; node 920; node 950]
+ siblings = [*; subtree 920; subtree 950]
},
 siblings = [node 20; *; node 80]
 }, * filled by node 50
+ siblings = [subtree 20; *; subtree 80]
+ }, * filled by subtree 50
We're understanding the `20` here in `node 20` to just be a metalanguage marker to help us theorists keep track of which node we're referring to. We're supposing the tree structure itself doesn't associate any informative labelling information with those nodes. It only associates informative labels with the tree leafs. (We haven't represented any such labels in our diagrams.)
We still do need to keep track of what fills the outermost targetted position`* filled by node 50`because that contain a subtree of arbitrary complexity, that is not determined by the rest of this data structure.
+We still do need to keep track of what fills the outermost targetted position`* filled by subtree 50`because that contain a subtree of arbitrary complexity, that is not determined by the rest of this data structure.
For simplicity, I'll continue to use the abbreviated form:
 {parent = ...; siblings = [node 20; *; node 80]}, * filled by node 50
+ {parent = ...; siblings = [subtree 20; *; subtree 80]}, * filled by subtree 50
But that should be understood as standing for the more fullyspelledout structure. Structures of this sort are called **tree zippers**, for a reason that will emerge. They should already seem intuitively similar to list zippers, though, at least in what we're using them to represent. I think it may initially be more helpful to call these **targetted trees**, though, and so will be switching back and forth between this different terms.
+But that should be understood as standing for the more fullyspelledout structure. Structures of this sort are called **tree zippers**. They should already seem intuitively similar to list zippers, at least in what we're using them to represent. I think it may also be helpful to call them **targetted trees**, though, and so will be switching back and forth between these different terms.
Moving left in our targetted tree that's targetted on `node 50` would be a matter of shifting the `*` leftwards:
 {parent = ...; siblings = [*; node 50; node 80]}, * filled by node 20
+ {parent = ...; siblings = [*; subtree 50; subtree 80]}, * filled by subtree 20
and similarly for moving right. If the sibling list is implemented as a list zipper, you should already know how to do that. If one were designing a tree zipper for a more restricted kind of tree, however, such as a binary tree, one would probably not represent siblings with a list zipper, but with something more specialpurpose and economical.
Moving downward in the tree would be a matter of constructing a tree targetted on some child of `node 20`, with the first part of the targetted tree above as its parent:
{
 parent = {parent = ...; siblings = [*; node 50; node 80]};
+ parent = {parent = ...; siblings = [*; subtree 50; subtree 80]};
siblings = [*; leaf 2; leaf 3]
}, * filled by leaf 1
How would we move upward in a tree? Well, we'd build a regular, untargetted tree with a root nodelet's call it `20`and whose children are given by the outermost sibling list in the targetted tree above, after inserting the targetted subtree into the `*` position:
+How would we move upward in a tree? Well, we'd build a regular, untargetted tree with a root nodelet's call it `20'`and whose children are given by the outermost sibling list in the targetted tree above, after inserting the targetted subtree into the `*` position:
 node 20
+ node 20'
/  \
/  \
leaf 1 leaf 2 leaf 3
We'll call this new untargetted tree `node 20`. The result of moving upward from our previous targetted tree, targetted on `leaf 1`, would be the outermost `parent` element of that targetted tree, with `node 20` being the subtree that fills that parent's target position `*`:
+We'll call this new untargetted tree `subtree 20'`. The result of moving upward from our previous targetted tree, targetted on `leaf 1`, would be the outermost `parent` element of that targetted tree, with `subtree 20'` being the subtree that fills that parent's target position `*`:
{
parent = ...;
 siblings = [*; node 50; node 80]
 }, * filled by node 20
+ siblings = [*; subtree 50; subtree 80]
+ }, * filled by subtree 20'
Or, spelling that structure out fully:
@@ 234,10 +220,10 @@ Or, spelling that structure out fully:
parent = None;
siblings = [*]
},
 siblings = [*; node 920; node 950]
+ siblings = [*; subtree 920; subtree 950]
},
 siblings = [*; node 50; node 80]
 }, * filled by node 20
+ siblings = [*; subtree 50; subtree 80]
+ }, * filled by subtree 20'
Moving upwards yet again would get us:
@@ 246,15 +232,15 @@ Moving upwards yet again would get us:
parent = None;
siblings = [*]
},
 siblings = [*; node 920; node 950]
 }, * filled by node 500
+ siblings = [*; subtree 920; subtree 950]
+ }, * filled by subtree 500'
where `node 500` refers to a tree built from a root node whose children are given by the list `[*; node 50; node 80]`, with `node 20` inserted into the `*` position. Moving upwards yet again would get us:
+where `subtree 500'` refers to a tree built from a root node whose children are given by the list `[*; subtree 50; subtree 80]`, with `subtree 20'` inserted into the `*` position. Moving upwards yet again would get us:
{
parent = None;
siblings = [*]
 }, * filled by node 9200
+ }, * filled by tree 9200'
where the targetted element is the root of our base tree. Like the "moving backward" operation for the list zipper, this "moving upward" operation is supposed to be reminiscent of closing a zipper, and that's why these data structures are called zippers.
@@ 266,7 +252,7 @@ We haven't given you a real implementation of the tree zipper, but only a sugges
##Samefringe using a tree zipper##
Recall back in [[Assignment4]], we asked you to enumerate the "fringe" of a leaflabeled tree. Both of these trees:
+Recall back in [[Assignment4]], we asked you to enumerate the "fringe" of a leaflabeled tree. Both of these trees (here I *am* drawing the labels in the diagram):
. .
/ \ / \
@@ 274,7 +260,7 @@ Recall back in [[Assignment4]], we asked you to enumerate the "fringe" of a leaf
/ \ / \
1 2 2 3
have the same fringe: `[1;2;3]`. We also asked you to write a function that determined when two trees have the same fringe. The way you approached that back then was to enumerate each tree's fringe, and then compare the two lists for equality. Today, and then again in a later class, we'll encounter two new ways to approach the problem of determining when two trees have the same fringe.
+have the same fringe: `[1;2;3]`. We also asked you to write a function that determined when two trees have the same fringe. The way you approached that back then was to enumerate each tree's fringe, and then compare the two lists for equality. Today, and then again in a later class, we'll encounter new ways to approach the problem of determining when two trees have the same fringe.
Supposing you did work out an implementation of the tree zipper, then one way to determine whether two trees have the same fringe would be: go downwards (and leftwards) in each tree as far as possible. Compare the targetted leaves. If they're different, stop because the trees have different fringes. If they're the same, then for each tree, move rightward if possible; if it's not (because you're at the rightmost position in a sibling list), more upwards then try again to move rightwards. Repeat until you are able to move rightwards. Once you do move rightwards, go downwards (and leftwards) as far as possible. Then you'll be targetted on the next leaf in the tree's fringe. The operations it takes to get to "the next leaf" may be different for the two trees. For example, in these trees:
@@ 338,9 +324,10 @@ Here is how you can extract the components of a labeled record:
Anyway, using record types, we might define the tree zipper interface like so:
 type 'a starred_tree = Root  Starring_Left of 'a starred_pair  Starring_Right of 'a starred_pair
 and 'a starred_pair = { parent : 'a starred_tree; sibling: 'a tree }
 and 'a zipper = { tree : 'a starred_tree; filler: 'a tree };;
+ type 'a starred_level = Root  Starring_Left of 'a starred_nonroot  Starring_Right of 'a starred_nonroot
+ and 'a starred_nonroot = { parent : 'a starred_level; sibling: 'a tree };;
+
+ type 'a zipper = { tree : 'a starred_level; filler: 'a tree };;
let rec move_botleft (z : 'a zipper) : 'a zipper =
(* returns z if the targetted node in z has no children *)