Use a type alias in binary tree exercise (#1589)

As suggested by @marshallpierce and @hurryabit

src/smart-pointers/exercise.md

@@ -11,6 +11,9 @@ nodes in N's right subtree will contain larger values.
 
 Implement the following types, so that the given tests pass.
 
+Extra Credit: implement an iterator over a binary tree that returns the values
+in order.
+
 ```rust,editable
 {{#include exercise.rs:types}}
 
@@ -18,6 +21,3 @@ Implement the following types, so that the given tests pass.
 
 {{#include exercise.rs:tests}}
 ```
-
-Extra Credit: implement an iterator over a binary tree that returns the values
-in order.

src/smart-pointers/exercise.rs

@@ -16,34 +16,55 @@
 use std::cmp::Ordering;
 
 // ANCHOR: types
+/// A node in the binary tree.
 #[derive(Debug)]
-struct BinaryTreeNode<T: Ord + Copy> {
+struct Node<T: Ord + Copy> {
     value: T,
-    left: BinaryTree<T>,
-    right: BinaryTree<T>,
+    left: Subtree<T>,
+    right: Subtree<T>,
 }
 
+/// A possibly-empty subtree.
+#[derive(Debug)]
+struct Subtree<T: Ord + Copy>(Option<Box<Node<T>>>);
+
 /// A container storing a set of values, using a binary tree.
 ///
 /// If the same value is added multiple times, it is only stored once.
 #[derive(Debug)]
-pub struct BinaryTree<T: Ord + Copy>(Option<Box<BinaryTreeNode<T>>>);
+pub struct BinaryTree<T: Ord + Copy> {
+    root: Subtree<T>,
+}
 // ANCHOR_END: types
 
 impl<T: Ord + Copy> BinaryTree<T> {
+    fn new() -> Self {
+        Self {
+            root: Subtree::new(),
+        }
+    }
+
+    fn insert(&mut self, value: T) {
+        self.root.insert(value);
+    }
+
+    fn has(&self, value: T) -> bool {
+        self.root.has(value)
+    }
+
+    fn len(&self) -> usize {
+        self.root.len()
+    }
+}
+
+impl<T: Ord + Copy> Subtree<T> {
     fn new() -> Self {
         Self(None)
     }
 
     fn insert(&mut self, value: T) {
         match &mut self.0 {
-            None => {
-                self.0 = Some(Box::new(BinaryTreeNode {
-                    value,
-                    left: BinaryTree::new(),
-                    right: BinaryTree::new(),
-                }));
-            }
+            None => self.0 = Some(Box::new(Node::new(value))),
             Some(n) => match value.cmp(&n.value) {
                 Ordering::Less => n.left.insert(value),
                 Ordering::Equal => {}
@@ -71,6 +92,16 @@ impl<T: Ord + Copy> BinaryTree<T> {
     }
 }
 
+impl<T: Ord + Copy> Node<T> {
+    fn new(value: T) -> Self {
+        Self {
+            value,
+            left: Subtree::new(),
+            right: Subtree::new(),
+        }
+    }
+}
+
 fn main() {
     let mut tree = BinaryTree::new();
     tree.insert("foo");