diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index f7b4fa1..bf4f97e 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -23,7 +23,7 @@ jobs:
       - uses: actions/checkout@v3
       # Build benchmarks to prevent bitrot
       - name: Build benchmarks
-        run: cargo build --workspace --benches
+        run: cargo build --workspace --benches --all-features
 
   doc-links:
     name: Intra-doc links
diff --git a/incrementalmerkletree/src/testing.rs b/incrementalmerkletree/src/testing.rs
index f511f51..c015cf8 100644
--- a/incrementalmerkletree/src/testing.rs
+++ b/incrementalmerkletree/src/testing.rs
@@ -65,17 +65,16 @@ pub trait Tree<H, C> {
 
     /// Creates a new checkpoint for the current tree state.
     ///
-    /// It is valid to have multiple checkpoints for the same tree state, and
-    /// each `rewind` call will remove a single checkpoint. Returns `false`
-    /// if the checkpoint identifier provided is less than or equal to the
-    /// maximum checkpoint identifier observed.
+    /// It is valid to have multiple checkpoints for the same tree state, and each `rewind` call
+    /// will remove a single checkpoint. Returns `false` if the checkpoint identifier provided is
+    /// less than or equal to the maximum checkpoint identifier observed.
     fn checkpoint(&mut self, id: C) -> bool;
 
-    /// Rewinds the tree state to the previous checkpoint, and then removes
-    /// that checkpoint record. If there are multiple checkpoints at a given
-    /// tree state, the tree state will not be altered until all checkpoints
-    /// at that tree state have been removed using `rewind`. This function
-    /// return false and leave the tree unmodified if no checkpoints exist.
+    /// Rewinds the tree state to the previous checkpoint, and then removes that checkpoint record.
+    ///
+    /// If there are multiple checkpoints at a given tree state, the tree state will not be altered
+    /// until all checkpoints at that tree state have been removed using `rewind`. This function
+    /// will return false and leave the tree unmodified if no checkpoints exist.
     fn rewind(&mut self) -> bool;
 }
 
@@ -288,7 +287,10 @@ pub fn check_operations<H: Hashable + Ord + Clone, C: Clone, T: Tree<H, C>>(
                         tree_checkpoints.push(tree_size);
                     }
                 } else {
-                    prop_assert_eq!(tree_size, 1 << tree.depth());
+                    prop_assert_eq!(
+                        tree_size,
+                        tree.current_position().map_or(0, |p| usize::from(p) + 1)
+                    );
                 }
             }
             CurrentPosition => {
@@ -375,7 +377,7 @@ pub fn compute_root_from_witness<H: Hashable>(value: H, position: Position, path
 // Types and utilities for cross-verification property tests
 //
 
-#[derive(Clone)]
+#[derive(Clone, Debug)]
 pub struct CombinedTree<H, C, I: Tree<H, C>, E: Tree<H, C>> {
     inefficient: I,
     efficient: E,
diff --git a/shardtree/Cargo.toml b/shardtree/Cargo.toml
index 53e36a0..e3f1c63 100644
--- a/shardtree/Cargo.toml
+++ b/shardtree/Cargo.toml
@@ -10,3 +10,27 @@ description = "A space-efficient Merkle tree with witnessing of marked leaves, c
 homepage = "https://github.com/zcash/incrementalmerkletree"
 repository = "https://github.com/zcash/incrementalmerkletree"
 categories = ["algorithms", "data-structures"]
+
+[dependencies]
+either = "1.8"
+incrementalmerkletree = { version = "0.3", path = "../incrementalmerkletree" }
+proptest = { version = "1.0.0", optional = true }
+
+[dev-dependencies]
+assert_matches = "1.5"
+criterion = "0.3"
+incrementalmerkletree = { version = "0.3", path = "../incrementalmerkletree", features = ["test-dependencies"] }
+proptest = "1.0.0"
+
+[features]
+test-dependencies = ["proptest"]
+
+[target.'cfg(unix)'.dev-dependencies]
+pprof = { version = "0.9", features = ["criterion", "flamegraph"] } # MSRV 1.56
+inferno = ">=0.11, <0.11.5" # MSRV 1.59
+
+[[bench]]
+name = "shardtree"
+harness = false
+required-features = ["test-dependencies"]
+
diff --git a/shardtree/benches/shardtree.rs b/shardtree/benches/shardtree.rs
new file mode 100644
index 0000000..ffba295
--- /dev/null
+++ b/shardtree/benches/shardtree.rs
@@ -0,0 +1,86 @@
+use criterion::{criterion_group, criterion_main, Criterion};
+use proptest::prelude::*;
+use proptest::strategy::ValueTree;
+use proptest::test_runner::TestRunner;
+
+use incrementalmerkletree::Address;
+use shardtree::{testing::arb_tree, Node};
+
+#[cfg(unix)]
+use pprof::criterion::{Output, PProfProfiler};
+
+// An algebra for computing the incomplete roots of a tree (the addresses at which nodes are
+// `Nil`). This is used for benchmarking to determine the viability of "attribute grammars" for
+// when you want to use `reduce` to compute a value that requires information to be passed top-down
+// through the tree.
+type RootFn = Box<dyn Fn(Address) -> Vec<Address>>;
+pub fn incomplete_roots<V: 'static>(node: Node<RootFn, V>) -> RootFn {
+    Box::new(move |addr| match &node {
+        Node::Parent { left, right, .. } => {
+            let (left_addr, right_addr) = addr
+                .children()
+                .expect("A parent node cannot appear at level 0");
+            let mut left_result = left(left_addr);
+            let mut right_result = right(right_addr);
+            left_result.append(&mut right_result);
+            left_result
+        }
+        Node::Leaf { .. } => vec![],
+        Node::Nil { .. } => vec![addr],
+    })
+}
+
+pub fn bench_shardtree(c: &mut Criterion) {
+    {
+        //let mut group = c.benchmark_group("shardtree-incomplete");
+
+        let mut runner = TestRunner::deterministic();
+        let input = arb_tree(Just(()), any::<String>(), 16, 4096)
+            .new_tree(&mut runner)
+            .unwrap()
+            .current();
+        println!(
+            "Benchmarking with {} leaves.",
+            input.reduce(
+                &(|node| match node {
+                    Node::Parent { left, right } => left + right,
+                    Node::Leaf { .. } => 1,
+                    Node::Nil => 0,
+                })
+            )
+        );
+
+        let input_root = Address::from_parts(
+            input
+                .reduce(
+                    &(|node| match node {
+                        Node::Parent { left, right } => std::cmp::max(left, right) + 1,
+                        Node::Leaf { .. } => 0,
+                        Node::Nil => 0,
+                    }),
+                )
+                .into(),
+            0,
+        );
+
+        c.bench_function("direct_recursion", |b| {
+            b.iter(|| input.incomplete(input_root))
+        });
+
+        c.bench_function("reduce", |b| {
+            b.iter(|| input.reduce(&incomplete_roots)(input_root))
+        });
+    }
+}
+
+#[cfg(unix)]
+criterion_group! {
+    name = benches;
+    config = Criterion::default().with_profiler(PProfProfiler::new(100, Output::Flamegraph(None)));
+    targets = bench_shardtree
+}
+
+#[cfg(not(unix))]
+criterion_group!(benches, bench_shardtree);
+
+criterion_main!(benches);
diff --git a/shardtree/src/lib.rs b/shardtree/src/lib.rs
index 8b13789..4986033 100644
--- a/shardtree/src/lib.rs
+++ b/shardtree/src/lib.rs
@@ -1 +1,242 @@
+use core::fmt::Debug;
+use core::ops::Deref;
+use either::Either;
+use std::rc::Rc;
 
+use incrementalmerkletree::Address;
+
+/// A "pattern functor" for a single layer of a binary tree.
+#[derive(Clone, Debug, PartialEq, Eq)]
+pub enum Node<C, A, V> {
+    /// A parent node in the tree, annotated with a value of type `A` and with left and right
+    /// children of type `C`.
+    Parent { ann: A, left: C, right: C },
+    /// A node of the tree that contains a value (usually a hash, sometimes with additional
+    /// metadata) and that has no children.
+    ///
+    /// Note that leaf nodes may appear at any position in the tree; i.e. they may contain computed
+    /// subtree root values and not just level-0 leaves.
+    Leaf { value: V },
+    /// The empty tree; a subtree or leaf for which no information is available.
+    Nil,
+}
+
+impl<C, A, V> Node<C, A, V> {
+    /// Returns whether or not this is the `Nil` tree.
+    ///
+    /// This is useful for cases where the compiler can automatically dereference an `Rc`, where
+    /// one would otherwise need additional ceremony to make an equality check.
+    pub fn is_nil(&self) -> bool {
+        matches!(self, Node::Nil)
+    }
+
+    /// Returns the contained leaf value, if this is a leaf node.
+    pub fn leaf_value(&self) -> Option<&V> {
+        match self {
+            Node::Parent { .. } => None,
+            Node::Leaf { value } => Some(value),
+            Node::Nil { .. } => None,
+        }
+    }
+
+    pub fn annotation(&self) -> Option<&A> {
+        match self {
+            Node::Parent { ann, .. } => Some(ann),
+            Node::Leaf { .. } => None,
+            Node::Nil => None,
+        }
+    }
+
+    /// Replaces the annotation on this node, if it is a `Node::Parent`; otherwise
+    /// returns this node unaltered.
+    pub fn reannotate(self, ann: A) -> Self {
+        match self {
+            Node::Parent { left, right, .. } => Node::Parent { ann, left, right },
+            other => other,
+        }
+    }
+}
+
+/// An F-algebra for use with [`Tree::reduce`] for determining whether a tree has any `Nil` nodes.
+///
+/// Returns `true` if no [`Node::Nil`] nodes are present in the tree.
+pub fn is_complete<A, V>(node: Node<bool, A, V>) -> bool {
+    match node {
+        Node::Parent { left, right, .. } => left && right,
+        Node::Leaf { .. } => true,
+        Node::Nil { .. } => false,
+    }
+}
+
+/// An immutable binary tree with each of its nodes tagged with an annotation value.
+#[derive(Clone, Debug, PartialEq, Eq)]
+pub struct Tree<A, V>(Node<Rc<Tree<A, V>>, A, V>);
+
+impl<A, V> Deref for Tree<A, V> {
+    type Target = Node<Rc<Tree<A, V>>, A, V>;
+    fn deref(&self) -> &Self::Target {
+        &self.0
+    }
+}
+
+impl<A, V> Tree<A, V> {
+    /// Replaces the annotation at the root of the tree, if the root is a `Node::Parent`; otherwise
+    /// returns this tree unaltered.
+    pub fn reannotate_root(self, ann: A) -> Tree<A, V> {
+        Tree(self.0.reannotate(ann))
+    }
+
+    /// Returns a vector of the addresses of [`Node::Nil`] subtree roots within this tree.
+    ///
+    /// The given address must correspond to the root of this tree, or this method will
+    /// yield incorrect results or may panic.
+    pub fn incomplete(&self, root_addr: Address) -> Vec<Address> {
+        match &self.0 {
+            Node::Parent { left, right, .. } => {
+                // We should never construct parent nodes where both children are Nil.
+                // While we could handle that here, if we encountered that case it would
+                // be indicative of a programming error elsewhere and so we assert instead.
+                assert!(!(left.0.is_nil() && right.0.is_nil()));
+                let (left_root, right_root) = root_addr
+                    .children()
+                    .expect("A parent node cannot appear at level 0");
+
+                let mut left_incomplete = left.incomplete(left_root);
+                let mut right_incomplete = right.incomplete(right_root);
+                left_incomplete.append(&mut right_incomplete);
+                left_incomplete
+            }
+            Node::Leaf { .. } => vec![],
+            Node::Nil => vec![root_addr],
+        }
+    }
+}
+
+impl<A: Clone, V: Clone> Tree<A, V> {
+    /// Folds over the tree from leaf to root with the given function.
+    ///
+    /// See [`is_complete`] for an example of a function that can be used with this method.
+    /// This operation will visit every node of the tree. See [`try_reduce`] for a variant
+    /// that can perform a depth-first, left-to-right traversal with the option to
+    /// short-circuit.
+    pub fn reduce<B, F: Fn(Node<B, A, V>) -> B>(&self, alg: &F) -> B {
+        match &self.0 {
+            Node::Parent { ann, left, right } => {
+                let left_result = left.reduce(alg);
+                let right_result = right.reduce(alg);
+                alg(Node::Parent {
+                    ann: ann.clone(),
+                    left: left_result,
+                    right: right_result,
+                })
+            }
+            Node::Leaf { value } => alg(Node::Leaf {
+                value: value.clone(),
+            }),
+            Node::Nil => alg(Node::Nil),
+        }
+    }
+
+    /// Folds over the tree from leaf to root with the given function.
+    ///
+    /// This performs a left-to-right, depth-first traversal that halts on the first
+    /// [`Either::Left`] result, or builds an [`Either::Right`] from the results computed at every
+    /// node.
+    pub fn try_reduce<L, R, F: Fn(Node<R, A, V>) -> Either<L, R>>(&self, alg: &F) -> Either<L, R> {
+        match &self.0 {
+            Node::Parent { ann, left, right } => left.try_reduce(alg).right_and_then(|l_value| {
+                right.try_reduce(alg).right_and_then(move |r_value| {
+                    alg(Node::Parent {
+                        ann: ann.clone(),
+                        left: l_value,
+                        right: r_value,
+                    })
+                })
+            }),
+            Node::Leaf { value } => alg(Node::Leaf {
+                value: value.clone(),
+            }),
+            Node::Nil => alg(Node::Nil),
+        }
+    }
+}
+
+#[cfg(any(bench, test, feature = "test-dependencies"))]
+pub mod testing {
+    use super::*;
+    use incrementalmerkletree::Hashable;
+    use proptest::prelude::*;
+
+    pub fn arb_tree<A: Strategy + Clone + 'static, V: Strategy + Clone + 'static>(
+        arb_annotation: A,
+        arb_leaf: V,
+        depth: u32,
+        size: u32,
+    ) -> impl Strategy<Value = Tree<A::Value, V::Value>>
+    where
+        A::Value: Clone + 'static,
+        V::Value: Hashable + Clone + 'static,
+    {
+        let leaf = prop_oneof![
+            Just(Tree(Node::Nil)),
+            arb_leaf.prop_map(|value| Tree(Node::Leaf { value }))
+        ];
+
+        leaf.prop_recursive(depth, size, 2, move |inner| {
+            (arb_annotation.clone(), inner.clone(), inner).prop_map(|(ann, left, right)| {
+                Tree(if left.is_nil() && right.is_nil() {
+                    Node::Nil
+                } else {
+                    Node::Parent {
+                        ann,
+                        left: Rc::new(left),
+                        right: Rc::new(right),
+                    }
+                })
+            })
+        })
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use crate::{Node, Tree};
+    use incrementalmerkletree::{Address, Level};
+    use std::rc::Rc;
+
+    #[test]
+    fn tree_incomplete() {
+        let t = Tree(Node::Parent {
+            ann: (),
+            left: Rc::new(Tree(Node::Nil)),
+            right: Rc::new(Tree(Node::Leaf { value: "a" })),
+        });
+        assert_eq!(
+            t.incomplete(Address::from_parts(Level::from(1), 0)),
+            vec![Address::from_parts(Level::from(0), 0)]
+        );
+
+        let t0 = Tree(Node::Parent {
+            ann: (),
+            left: Rc::new(Tree(Node::Leaf { value: "b" })),
+            right: Rc::new(t.clone()),
+        });
+        assert_eq!(
+            t0.incomplete(Address::from_parts(Level::from(2), 1)),
+            vec![Address::from_parts(Level::from(0), 6)]
+        );
+
+        let t1 = Tree(Node::Parent {
+            ann: (),
+            left: Rc::new(Tree(Node::Nil)),
+            right: Rc::new(t),
+        });
+        assert_eq!(
+            t1.incomplete(Address::from_parts(Level::from(2), 1)),
+            vec![
+                Address::from_parts(Level::from(1), 2),
+                Address::from_parts(Level::from(0), 6)
+            ]
+        );
+    }
+}