diff --git a/src/merkle/index.rs b/src/merkle/index.rs
index e9d5e0e..cedc435 100644
--- a/src/merkle/index.rs
+++ b/src/merkle/index.rs
@@ -83,20 +83,29 @@ impl NodeIndex {
     /// # Panics
     /// Panics if the depth indicated by `index` does not fit in a [`u8`], or if the row-value
     /// indicated by `index` does not fit in a [`u64`].
-    pub const fn from_scalar_index(index: NonZero<u128>) -> Result<Self, MerkleError> {
+    pub fn from_scalar_index(index: NonZero<u128>) -> Result<Self, MerkleError> {
         let index = index.get() - 1;
 
         if index == 0 {
-            return Ok(Self { depth: 0, value: 0 });
+            return Ok(Self::root());
+        }
+
+        // The log of 1 is always 0.
+        if index == 1 {
+            return Ok(Self::root().left_child());
         }
 
         let depth = {
-            let depth = u128::ilog2(index);
+            let depth = u128::ilog2(index + 1);
             assert!(depth <= u8::MAX as u32);
             depth as u8
         };
 
         let max_value_for_depth = (1 << depth) - 1;
+        assert!(
+            max_value_for_depth <= u64::MAX as u128,
+            "max_value ({max_value_for_depth}) does not fit in u64",
+        );
 
         let value = {
             let value = index - max_value_for_depth;
@@ -125,6 +134,18 @@ impl NodeIndex {
         self
     }
 
+    pub const fn left_ancestor_n(mut self, n: u8) -> Self {
+        self.depth += n;
+        self.value <<= n;
+        self
+    }
+
+    pub const fn right_ancestor_n(mut self, n: u8) -> Self {
+        self.depth += n;
+        self.value = (self.value << n) + 1;
+        self
+    }
+
     /// Returns right child index of the current node.
     pub const fn right_child(mut self) -> Self {
         self.depth += 1;
@@ -132,6 +153,68 @@ impl NodeIndex {
         self
     }
 
+    /// Returns the parent of the current node.
+    pub const fn parent(mut self) -> Self {
+        self.depth = self.depth.saturating_sub(1);
+        self.value >>= 1;
+        self
+    }
+
+    /// Returns the `n`th parent of the current node.
+    pub fn parent_n(mut self, n: u8) -> Self {
+        debug_assert!(n <= self.depth);
+        self.depth = self.depth.saturating_sub(n);
+        self.value >>= n;
+
+        self
+    }
+
+    /// Returns `true` if and only if `other` is an ancestor of the current node, or the current
+    /// node itself.
+    pub fn contains(&self, mut other: Self) -> bool {
+        if other == *self {
+            return true;
+        }
+        if other.is_root() {
+            return false;
+        }
+        if other.depth < self.depth {
+            return false;
+        }
+
+        other = other.parent_n(other.depth() - self.depth());
+
+        loop {
+            if other == *self {
+                return true;
+            }
+
+            if other.is_root() {
+                return false;
+            }
+
+            other = other.parent();
+        }
+    }
+
+    /// Returns the right-most descendent of the current node for a tree of `DEPTH` depth.
+    pub const fn rightmost_descendent<const DEPTH: u8>(mut self) -> Self {
+        while self.depth() < DEPTH {
+            self = self.right_child();
+        }
+
+        self
+    }
+
+    /// Returns the left-most descendent of the current node for a tree of `DEPTH` depth.
+    pub const fn leftmost_descendent<const DEPTH: u8>(mut self) -> Self {
+        while self.depth() < DEPTH {
+            self = self.left_child();
+        }
+
+        self
+    }
+
     // PROVIDERS
     // --------------------------------------------------------------------------------------------
 
@@ -245,6 +328,21 @@ mod tests {
         assert!(NodeIndex::new(64, u64::MAX).is_ok());
     }
 
+    #[test]
+    fn test_scalar_roundtrip() {
+        // Arbitrary value that's at the bottom and not in a corner.
+        let start = NodeIndex::make(64, u64::MAX - 8);
+
+        let mut index = start;
+        while !index.is_root() {
+            let as_scalar = index.to_scalar_index();
+            let round_trip =
+                NodeIndex::from_scalar_index(NonZero::new(as_scalar).unwrap()).unwrap();
+            assert_eq!(index, round_trip, "{index:?} did not round-trip as a scalar index");
+            index.move_up();
+        }
+    }
+
     prop_compose! {
         fn node_index()(value in 0..2u64.pow(u64::BITS - 1)) -> NodeIndex {
             // unwrap never panics because the range of depth is 0..u64::BITS