+#Implementing trees#
+
+In [[Assignment3]] we proposed a very ad-hoc-ish implementation of trees.
+
+Think about how you'd implement them in a more principled way. You could
+use any of the version 1 -- version 5 implementation of lists as a model.
+
+To keep things simple, we'll stick to binary trees. A node will either be a
+*leaf* of the tree, or it will have exactly two children.
+
+There are two kinds of trees to think about. In one sort of tree, it's only
+the tree's leaves that are labeled:
+
+ .
+ / \
+ . 3
+ / \
+ 1 2
+
+Linguists often use trees of this sort. The inner, non-leaf nodes of the
+tree do have associated values. But what values they are can be determined from
+the structure of the tree and the values of the node's left and right children.
+So the inner node doesn't need its own independent label.
+
+In another sort of tree, the tree's inner nodes are also labeled:
+
+ 4
+ / \
+ 2 5
+ / \
+ 1 3
+
+When you want to efficiently arrange an ordered collection, so that it's
+easy to do a binary search through it, this is the way you usually structure
+your data.
+
+These latter sorts of trees can helpfully be thought of as ones where
+*only* the inner nodes are labeled. Leaves can be thought of as special,
+dead-end branches with no label:
+
+ .4.
+ / \
+ 2 5
+ / \ / \
+ 1 3 x x
+ / \ / \
+ x x x x
+
+In our earlier discussion of lists, we said they could be thought of as
+data structures of the form:
+
+ Empty_list | Non_empty_list (its_head, its_tail)
+
+And that could in turn be implemented in v2 form as:
+
+ the_list (\head tail. non_empty_handler) empty_handler
+
+Similarly, the leaf-labeled tree:
+
+ .
+ / \
+ . 3
+ / \
+ 1 2
+
+can be thought of as a data structure of the form:
+
+ Leaf (its_label) | Non_leaf (its_left_subtree, its_right_subtree)
+
+and that could be implemented in v2 form as:
+
+ the_tree (\left right. non_leaf_handler) (\label. leaf_handler)
+
+And the node-labeled tree:
+
+ .4.
+ / \
+ 2 5
+ / \ / \
+ 1 3 x x
+ / \ / \
+ x x x x
+
+can be thought of as a data structure of the form:
+
+ Leaf | Non_leaf (its_left_subtree, its_label, its_right_subtree)
+
+and that could be implemented in v2 form as:
+
+ the_tree (\left label right. non_leaf_handler) leaf_result
+
+
+What would correspond to "folding" a function `f` and base value `z` over a
+tree? Well, if it's an empty tree:
+
+ x
+
+we should presumably get back `z`. And if it's a simple, non-empty tree:
+
+ 1
+ / \
+ x x
+
+we should expect something like `f z 1 z`, or `f <result of folding f and z
+over left subtree> label_of_this_node <result of folding f and z over right
+subtree>`. (It's not important what order we say `f` has to take its arguments
+in.)
+
+A v3-style implementation of node-labeled trees, then, might be:
+
+ let empty_tree = \f z. z in
+ let make_tree = \left label right. \f z. f (left f z) label (right f z) in
+ ...
+
+Think about how you might implement other tree operations, such as getting
+the label of the root (topmost node) of a tree; extracting the left subtree of
+a node; and so on.
+
+Think about different ways you might implement leaf-labeled trees.
+
+If you had one tree and wanted to make a larger tree out of it, adding in a
+new element, how would you do that?
+
+When using trees to represent linguistic structures, one doesn't have
+latitude about *how* to build a larger tree. The linguistic structure you're
+trying to represent will determine where the new element should be placed, and
+where the previous tree should be placed.
+
+However, when using trees as a computational tool, one usually does have
+latitude about how to structure a larger tree---in the same way that we had the
+freedom to implement our sets with lists whose members were just appended in
+the order we built the set up, or instead with lists whose members were ordered
+numerically.
+
+When building a new tree, one strategy for where to put the new element and
+where to put the existing tree would be to always lean towards a certain side.
+For instance, to add the element `2` to the tree:
+
+ 1
+ / \
+ x x
+
+we might construct the following tree:
+
+ 1
+ / \
+ x 2
+ / \
+ x x
+
+or perhaps we'd do it like this instead:
+
+ 2
+ / \
+ x 1
+ / \
+ x x
+
+However, if we always leaned to the right side in this way, then the tree
+would get deeper and deeper on that side, but never on the left:
+
+ 1
+ / \
+ x 2
+ / \
+ x 3
+ / \
+ x 4
+ / \
+ x 5
+ / \
+ x x
+
+and that wouldn't be so useful if you were using the tree as an arrangement
+to enable *binary searches* over the elements it holds. For that, you'd prefer
+the tree to be relatively "balanced", like this:
+
+ .4.
+ / \
+ 2 5
+ / \ / \
+ 1 3 x x
+ / \ / \
+ x x x x
+
+Do you have any ideas about how you might efficiently keep the new trees
+you're building pretty "balanced" in this way?
+
+This is a large topic in computer science. There's no need for you to learn
+the various strategies that they've developed for doing this. But
+thinking in broad brush-strokes about what strategies might be promising will
+help strengthen your understanding of trees, and useful ways to implement them
+in a purely functional setting like the lambda calculus.
+