diff --git a/.gitignore b/.gitignore
index 9386f3b..d97d817 100644
--- a/.gitignore
+++ b/.gitignore
@@ -2,3 +2,4 @@
 /target
 /Cargo.lock
 **/*.rs.bk
+metrics.ansi
diff --git a/src/flow/binary/balancing.rs b/src/flow/binary/balancing.rs
index 2d8165a..721dc5b 100644
--- a/src/flow/binary/balancing.rs
+++ b/src/flow/binary/balancing.rs
@@ -1,4 +1,6 @@
-use super::*;
+use std::fmt::Display;
+
+use super::{avl::*, bounds::bound::BinaryTreesBindable, *};
 
 pub trait BinaryTreesUnbalanced<'a>: BinaryTreesHeight<'a> {
     fn tree_of_with_height(&self, node: Self::Node, height: u64) -> BTWrap<'a, Self, Self::Tree>;
@@ -25,6 +27,12 @@ pub trait BinaryTreesUnbalanced<'a>: BinaryTreesHeight<'a> {
 #[derive(Clone)]
 pub struct BalancedTrees<BT>(BT);
 
+impl<BT> BalancedTrees<BT> {
+    pub fn new(bt: BT) -> Self {
+        Self(bt)
+    }
+}
+
 impl<'a, BT: FunctorContext<'a>> FunctorContext<'a> for BalancedTrees<BT> {
     type T = BT::T;
 }
@@ -37,6 +45,25 @@ pub enum BalancingError {
     HeightMismatch { children: (u64, u64), parent: u64 },
 }
 
+impl Display for BalancingError {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        match self {
+            Self::NodeHeight => write!(f, "invalid AVL non-leaf height: 0."),
+            Self::LeafHeight(height) => {
+                write!(f, "invalid AVL leaf height: {height}!=0.")
+            }
+            Self::Balance(hl, hr) => write!(f, "unbalanced AVL node: {hl} {hr}."),
+            Self::HeightOverflow => write!(f, "AVL tree height overflow"),
+            Self::HeightMismatch { children, parent } => {
+                write!(
+                    f,
+                    "AVL child-parent height mismatch: {children:?}, {parent}"
+                )
+            }
+        }
+    }
+}
+
 fn balanced(hl: u64, hr: u64) -> Result<(), BalancingError> {
     if hl.abs_diff(hr) > 1 {
         Err(BalancingError::Balance(hl, hr))
@@ -151,3 +178,24 @@ impl<'a, BT: BinaryTreesUnbalanced<'a> + BinaryTreesTryJoin<'a>> BinaryTreesTryJ
             .balancing_bind(balanced(hl, hr), |_| self.0.try_join(tl, key, tr))
     }
 }
+
+impl<'a, BT: BinaryTreesUnbalanced<'a> + BinaryTreesBindable<'a>> BinaryTreesBindable<'a>
+    for BalancedTrees<BT>
+{
+    fn bounds_error<T: 'a>(&self, error: bounds::BoundsError<Self::Key>) -> BTWrap<'a, Self, T> {
+        self.0.bounds_error(error)
+    }
+}
+
+impl<'a, BT: BinaryTreesUnbalanced<'a> + BinaryTreesEmpty<'a> + BinaryTreesTryJoin<'a>>
+    BinaryTreesMutable<'a> for BalancedTrees<BT>
+{
+    fn join_key(
+        self,
+        tl: Self::Tree,
+        key: Self::Key,
+        tr: Self::Tree,
+    ) -> BTWrap<'a, Self, Self::Node> {
+        self.join_key_balanced(tl, key, tr)
+    }
+}
diff --git a/src/mrds/trees.rs b/src/mrds/trees.rs
index 3380f2e..d1866cf 100644
--- a/src/mrds/trees.rs
+++ b/src/mrds/trees.rs
@@ -1,2 +1,3 @@
 pub mod avl;
+pub mod heighted;
 pub mod unbalanced;
diff --git a/src/mrds/trees/avl.rs b/src/mrds/trees/avl.rs
index 90fc8f3..bd187c5 100644
--- a/src/mrds/trees/avl.rs
+++ b/src/mrds/trees/avl.rs
@@ -1,7 +1,7 @@
 use std::{fmt::Display, marker::PhantomData, rc::Rc};
 
 use crate::flow::{
-    binary::{avl::*, bounds::bound::BinaryTreesBindable, *},
+    binary::{avl::*, bounds::bound::*, *},
     comparator::*,
 };
 use crate::func::{context::*, *};
@@ -199,14 +199,12 @@ impl<'a, A: 'a + PartialOrd + Clone> BinaryTreesTryJoin<'a> for AvlTs<A> {
 
 impl<'a, A: 'a + PartialOrd + Clone> BinaryTreesBindable<'a> for AvlTs<A> {
     fn bounds_error<T: 'a>(&self, _error: bounds::BoundsError<Self::Key>) -> BTWrap<'a, Self, T> {
-        panic!("bounds violated");
+        panic!("bounds violated.");
     }
 }
 
 #[cfg(test)]
 mod tests {
-    use crate::flow::binary::bounds::bound::*;
-
     use super::*;
 
     #[test]
diff --git a/src/mrds/trees/heighted.rs b/src/mrds/trees/heighted.rs
new file mode 100644
index 0000000..1201006
--- /dev/null
+++ b/src/mrds/trees/heighted.rs
@@ -0,0 +1,245 @@
+use std::{fmt::Display, marker::PhantomData, rc::Rc};
+
+use crate::flow::{
+    binary::{balancing::*, bounds::bound::*, *},
+    comparator::*,
+};
+use crate::func::{context::*, *};
+
+#[derive(Clone)]
+struct Node<A> {
+    l: Tree<A>,
+    r: Tree<A>,
+    key: A,
+}
+
+impl<A: Display> Display for Node<A> {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        write!(f, "({} {} {})", self.l, self.key, self.r)
+    }
+}
+
+struct Reference<A> {
+    node: Rc<Node<A>>,
+}
+
+impl<A> Clone for Reference<A> {
+    fn clone(&self) -> Self {
+        Self {
+            node: self.node.clone(),
+        }
+    }
+}
+
+impl<A: Display> Display for Reference<A> {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        write!(f, "{}", self.node)
+    }
+}
+
+struct Tree<A> {
+    reference: Option<Reference<A>>,
+    height: u64,
+}
+
+impl<A> Clone for Tree<A> {
+    fn clone(&self) -> Self {
+        Self {
+            reference: self.reference.clone(),
+            height: self.height,
+        }
+    }
+}
+
+impl<A: Display> Display for Tree<A> {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        match &self.reference {
+            Some(reference) => write!(f, "{}", reference),
+            None => write!(f, "-"),
+        }
+    }
+}
+
+struct Trees<A> {
+    _a: PhantomData<A>,
+}
+
+impl<A> Trees<A> {
+    fn new() -> Self {
+        Self { _a: PhantomData }
+    }
+}
+
+impl<A> Clone for Trees<A> {
+    fn clone(&self) -> Self {
+        Self::new()
+    }
+}
+
+impl<'a, A: 'a> FunctorContext<'a> for Trees<A> {
+    type T = instances::solo::SoloInstance;
+}
+
+impl<'a, A: 'a + PartialOrd + Clone> BinaryTrees<'a> for Trees<A> {
+    type Node = Node<A>;
+
+    type Reference = Reference<A>;
+
+    type Tree = Tree<A>;
+
+    type Key = A;
+
+    type Comparator = DefaultComparator;
+
+    type _Tm = Self::T;
+
+    fn comparator(&self) -> &Self::Comparator {
+        &DefaultComparator
+    }
+
+    fn split(&self, node: &Self::Node) -> Split<'a, Self> {
+        (node.l.clone(), node.r.clone(), node.key.clone())
+    }
+
+    fn resolve(&self, reference: &Self::Reference) -> BTWrap<'a, Self, Self::Node> {
+        reference.node.as_ref().clone()
+    }
+
+    fn equal(&self, _rhs: &Self::Reference, _lhs: &Self::Reference) -> bool {
+        false
+    }
+
+    fn refer(&self, tree: &Self::Tree) -> Option<Self::Reference> {
+        tree.reference.clone()
+    }
+}
+
+impl<'a, A: 'a + PartialOrd + Clone> BinaryTreesHeight<'a> for Trees<A> {
+    fn height(&self, tree: &Self::Tree) -> u64 {
+        tree.height
+    }
+
+    fn leaf_height_error<T: 'a>(&self, height: u64) -> BTWrap<'a, Self, T> {
+        panic!("leaf height error: {height}.")
+    }
+}
+
+impl<'a, A: 'a + PartialOrd + Clone> BinaryTreesEmpty<'a> for Trees<A> {
+    fn empty(&self) -> Self::Tree {
+        Tree {
+            reference: None,
+            height: 0,
+        }
+    }
+
+    fn split_key_empty(
+        &self,
+        _tree: Self::Tree,
+        _key: Self::Key,
+    ) -> BTWrap<'a, Self, KeySplit<'a, Self>> {
+        (self.empty(), self.empty())
+    }
+}
+
+impl<'a, A: 'a + PartialOrd + Clone> BinaryTreesTryJoin<'a> for Trees<A> {
+    fn try_join(
+        &self,
+        tl: Self::Tree,
+        key: Self::Key,
+        tr: Self::Tree,
+    ) -> BTWrap<'a, Self, Self::Node> {
+        Node { l: tl, r: tr, key }
+    }
+}
+
+impl<'a, A: 'a + PartialOrd + Clone> BinaryTreesUnbalanced<'a> for Trees<A> {
+    fn tree_of_with_height(&self, node: Self::Node, height: u64) -> BTWrap<'a, Self, Self::Tree> {
+        Tree {
+            reference: Some(Reference {
+                node: Rc::new(node),
+            }),
+            height,
+        }
+    }
+
+    fn balancing_error<T: 'a>(&self, _error: BalancingError) -> BTWrap<'a, Self, T> {
+        panic!("balancing error.")
+    }
+}
+
+impl<'a, A: 'a + PartialOrd + Clone> BinaryTreesBindable<'a> for Trees<A> {
+    fn bounds_error<T: 'a>(&self, _error: bounds::BoundsError<Self::Key>) -> BTWrap<'a, Self, T> {
+        panic!("bounds violated.");
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    impl<A> Node<A> {
+        fn balanced(&self) -> bool {
+            let (hl, hr) = (self.l.height, self.r.height);
+            std::cmp::max(hl, hr) - std::cmp::min(hl, hr) < 2
+                && self.l.balanced()
+                && self.r.balanced()
+        }
+    }
+
+    impl<A> Reference<A> {
+        fn balanced(&self) -> bool {
+            self.node.balanced()
+        }
+    }
+
+    impl<A> Tree<A> {
+        fn balanced(&self) -> bool {
+            match &self.reference {
+                Some(reference) => reference.balanced(),
+                None => true,
+            }
+        }
+    }
+
+    #[test]
+    fn test() {
+        let trees = BalancedTrees::new(Trees::new());
+        let mut tree = trees.empty();
+        for i in [
+            8, 3, 10, 17, 0, 13, 6, 1, 11, 5, 4, 7, 18, 14, 15, 9, 2, 19, 16, 12,
+        ] {
+            tree = trees.clone().add_tree(tree, i);
+            assert!(tree.balanced());
+            // println!("{} {}", tree.height, tree);
+        }
+        for i in [
+            2, 9, 4, 7, 8, 10, 17, 1, 13, 15, 18, 12, 5, 0, 3, 6, 16, 19, 14, 11,
+        ] {
+            tree = trees.clone().remove(tree, i);
+            assert!(tree.balanced());
+            // println!("{} {}", tree.height, tree);
+        }
+        // assert!(false);
+    }
+
+    #[test]
+    fn test_with_bounds() {
+        let trees = BoundTrees::new(BalancedTrees::new(Trees::new()));
+        let mut tree = trees.empty();
+        for i in [
+            8, 3, 10, 17, 0, 13, 6, 1, 11, 5, 4, 7, 18, 14, 15, 9, 2, 19, 16, 12,
+        ] {
+            tree = trees.clone().add_tree(tree, i);
+            assert!(tree.balanced());
+            // println!("{} {}", tree.height, tree);
+        }
+        for i in [
+            2, 9, 4, 7, 8, 10, 17, 1, 13, 15, 18, 12, 5, 0, 3, 6, 16, 19, 14, 11,
+        ] {
+            tree = trees.clone().remove(tree, i);
+            assert!(tree.balanced());
+            // println!("{} {}", tree.height, tree);
+        }
+        // assert!(false);
+    }
+}