diff --git a/tensorflow_probability/python/experimental/autobnn/bnn_tree.py b/tensorflow_probability/python/experimental/autobnn/bnn_tree.py
index 4a02f23252..18e7dacaf2 100644
--- a/tensorflow_probability/python/experimental/autobnn/bnn_tree.py
+++ b/tensorflow_probability/python/experimental/autobnn/bnn_tree.py
@@ -30,6 +30,7 @@
 LEAVES = [
     kernels.ExponentiatedQuadraticBNN,
     kernels.MaternBNN,
+    kernels.ExponentialBNN,
     kernels.LinearBNN,
     kernels.QuadraticBNN,
     kernels.PeriodicBNN,
@@ -49,6 +50,7 @@
 NON_PERIODIC_KERNELS = [
     kernels.ExponentiatedQuadraticBNN,
     kernels.MaternBNN,
+    kernels.ExponentialBNN,
     kernels.LinearBNN,
     kernels.QuadraticBNN,
     kernels.OneLayerBNN,
@@ -86,7 +88,7 @@ def list_of_all(
   # Abelian operators that aren't Multiply.
   if include_sums:
     for i, c1 in enumerate(non_multiply_children):
-      for j in range(i + 1):
+      for j in range(i):
         c2 = non_multiply_children[j]
         # Add is also abelian, but WeightedSum is more general.
         all_bnns.append(
diff --git a/tensorflow_probability/python/experimental/autobnn/bnn_tree_test.py b/tensorflow_probability/python/experimental/autobnn/bnn_tree_test.py
index 10b38b24c2..67e6f87334 100644
--- a/tensorflow_probability/python/experimental/autobnn/bnn_tree_test.py
+++ b/tensorflow_probability/python/experimental/autobnn/bnn_tree_test.py
@@ -25,24 +25,25 @@
 
 class TreeTest(parameterized.TestCase):
 
-  def test_list_of_all(self):
+  def test_list_of_all_depth0(self):
     l0 = bnn_tree.list_of_all(jnp.linspace(0.0, 100.0, 100), 0)
-    # With no periods, there should be five kernels.
-    self.assertLen(l0, 5)
+    # With no periods, there should be six kernels.
+    self.assertLen(l0, 6)
     for k in l0:
       self.assertFalse(k.going_to_be_multiplied)
 
     l0 = bnn_tree.list_of_all(100, 0, 50, [20.0, 40.0], parent_is_multiply=True)
-    self.assertLen(l0, 7)
+    self.assertLen(l0, 8)
     for k in l0:
       self.assertTrue(k.going_to_be_multiplied)
 
+  def test_list_of_all_depth1(self):
     l1 = bnn_tree.list_of_all(jnp.linspace(0.0, 100.0, 100), 1)
     # With no periods, there should be
-    # 15 trees with a Multiply top node,
-    # 15 trees with a WeightedSum top node, and
-    # 25 trees with a LearnableChangePoint top node.
-    self.assertLen(l1, 55)
+    # choose(6+1, 2) = 21 trees with a Multiply top node,
+    # choose(6, 2) = 15 trees with a WeightedSum top node, and
+    # 6*6 = 36 trees with a LearnableChangePoint top node.
+    self.assertLen(l1, 72)
 
     # Check that all of the BNNs in the tree can be trained.
     for k in l1:
@@ -63,12 +64,13 @@ def test_list_of_all(self):
     # nodes, with 7*8/2 = 28 trees.
     self.assertLen(l1, 28)
 
+  def test_list_of_all_depth2(self):
     l2 = bnn_tree.list_of_all(jnp.linspace(0.0, 100.0, 100), 2)
-    # With no periods, there should be
-    # 15*16/2 = 120 trees with a Multiply top node,
-    # 55*56/2 = 1540 trees with a WeightedSum top node, and
-    # 55*55 = 3025 trees with a LearnableChangePoint top node.
-    self.assertLen(l2, 4685)
+    # There are 66 trees of depth 1, of which 15 are safe to multiply.
+    # choose(15+1, 2) = 120 trees with a Multiply top node,
+    # choose(66, 2) = 2145 trees with a WeightedSum top node, and
+    # 66*66 = 4356 trees with a LearnableChangePoint top node.
+    self.assertLen(l2, 7860)
 
   @parameterized.parameters(0, 1)  # depth=2 segfaults on my desktop :(
   def test_weighted_sum_of_all(self, depth):
diff --git a/tensorflow_probability/python/experimental/autobnn/kernels.py b/tensorflow_probability/python/experimental/autobnn/kernels.py
index b02ffb7316..f84dca7f4e 100644
--- a/tensorflow_probability/python/experimental/autobnn/kernels.py
+++ b/tensorflow_probability/python/experimental/autobnn/kernels.py
@@ -162,11 +162,25 @@ def kernel_init(seed, shape, unused_dtype):
     self.kernel_init = kernel_init
     super().setup()
 
+  def distributions(self):
+    d = super().distributions()
+    d['dense1']['kernel'] = student_t_lib.StudentT(
+        df=2.0 * self.degrees_of_freedom, loc=0.0, scale=1.0)
+    return d
+
   def summarize(self, params=None, full: bool = False) -> str:
     """Return a string summarizing the structure of the BNN."""
     return f'{self.shortname()}({self.degrees_of_freedom})'
 
 
+class ExponentialBNN(MaternBNN):
+  """Matern(0.5), also known as the absolute exponential kernel."""
+  degrees_of_freedom: float = 0.5
+
+  def summarize(self, params=None, full: bool = False) -> str:
+    return self.shortname()
+
+
 class PolynomialBNN(OneLayerBNN):
   """A BNN where samples are polynomial functions."""
   degree: int = 2
diff --git a/tensorflow_probability/python/experimental/autobnn/kernels_test.py b/tensorflow_probability/python/experimental/autobnn/kernels_test.py
index 67e574d517..c5f26882eb 100644
--- a/tensorflow_probability/python/experimental/autobnn/kernels_test.py
+++ b/tensorflow_probability/python/experimental/autobnn/kernels_test.py
@@ -30,6 +30,7 @@
     kernels.OneLayerBNN,
     kernels.ExponentiatedQuadraticBNN,
     kernels.MaternBNN,
+    kernels.ExponentialBNN,
     kernels.PeriodicBNN,
     kernels.PolynomialBNN,
     kernels.LinearBNN,
@@ -139,6 +140,7 @@ def test_likelihood(self, kernel):
       (kernels.OneLayerBNN(width=10), 'OneLayer'),
       (kernels.ExponentiatedQuadraticBNN(width=5), 'RBF'),
       (kernels.MaternBNN(width=5), 'Matern(2.5)'),
+      (kernels.ExponentialBNN(width=20), 'Exponential'),
       (kernels.PeriodicBNN(period=10, width=10), 'Periodic(period=10.00)'),
       (kernels.PolynomialBNN(degree=3, width=2), 'Polynomial(degree=3)'),
       (kernels.LinearBNN(width=5), 'Linear'),