diff --git a/python/tests/test_haplotype_matching.py b/python/tests/test_haplotype_matching.py
index bb98d5bfd3..353fc5db93 100644
--- a/python/tests/test_haplotype_matching.py
+++ b/python/tests/test_haplotype_matching.py
@@ -336,6 +336,7 @@ def update_probabilities(self, site, haplotype_state):
                 match = (
                     haplotype_state == MISSING or haplotype_state == allelic_state[v]
                 )
+                # Note that the node u is used only by Viterbi
                 st.value = self.compute_next_probability(site.id, st.value, match, u)
 
         # Unset the states
@@ -383,12 +384,15 @@ def compute_emission_proba(self, site_id, is_match):
                     p_e = mu / (n_alleles - 1)
         return p_e
 
-    def run(self, h):
-        n = self.ts.num_samples
+    def initialise(self, value):
         self.tree.clear()
         for u in self.ts.samples():
             self.T_index[u] = len(self.T)
-            self.T.append(ValueTransition(tree_node=u, value=1 / n))
+            self.T.append(ValueTransition(tree_node=u, value=value))
+
+    def run(self, h):
+        n = self.ts.num_samples
+        self.initialise(1 / n)
         while self.tree.next():
             self.update_tree()
             for site in self.tree.sites():
@@ -533,7 +537,9 @@ def traceback(self):
 
 
 class ForwardAlgorithm(LsHmmAlgorithm):
-    """Runs the Li and Stephens forward algorithm."""
+    """
+    The Li and Stephens forward algorithm.
+    """
 
     def __init__(
         self, ts, rho, mu, alleles, n_alleles, scale_mutation=False, precision=10
@@ -557,7 +563,6 @@ def compute_normalisation_factor(self):
             s += self.N[j] * st.value
         return s
 
-    # Note node only used in Viterbi
     def compute_next_probability(self, site_id, p_last, is_match, node):
         rho = self.rho[site_id]
         n = self.ts.num_samples
@@ -566,8 +571,10 @@ def compute_next_probability(self, site_id, p_last, is_match, node):
         return p_t * p_e
 
 
-class MirroredBackwardAlgorithm(LsHmmAlgorithm):
-    """Runs the Li and Stephens backward algorithm."""
+class BackwardAlgorithm(ForwardAlgorithm):
+    """
+    The Li and Stephens backward algorithm.
+    """
 
     def __init__(
         self,
@@ -591,50 +598,38 @@ def __init__(
         )
         self.output = CompressedMatrix(ts, normalisation_factor)
 
-    def compute_normalisation_factor(self):
-        s = 0
-        for j, st in enumerate(self.T):
-            assert st.tree_node != tskit.NULL
-            assert self.N[j] > 0
-            s += self.N[j] * st.value
-        return s
-
     def compute_next_probability(self, site_id, p_next, is_match, node):
         p_e = self.compute_emission_proba(site_id, is_match)
         return p_next * p_e
 
     def process_site(self, site, haplotype_state):
+        s = self.output.normalisation_factor[site.id]
         self.output.store_site(
             site.id,
-            self.output.normalisation_factor[site.id],
+            s,
             [(st.tree_node, st.value) for st in self.T],
         )
         self.update_probabilities(site, haplotype_state)
         self.compress()
         b_last_sum = self.compute_normalisation_factor()
-        s = self.output.normalisation_factor[site.id]
+        n = self.ts.num_samples
+        rho = self.rho[site.id]
         for st in self.T:
             if st.tree_node != tskit.NULL:
-                st.value = (self.rho[site.id] / self.ts.num_samples) * b_last_sum + (
-                    1 - self.rho[site.id]
-                ) * st.value
+                st.value = rho * b_last_sum / n + (1 - rho) * st.value
                 st.value /= s
                 st.value = round(st.value, self.precision)
 
     def run(self, h):
-        # NOTE value=1 is the difference to the standard code for fwd and vit
-        self.tree.clear()
-        for u in self.ts.samples():
-            self.T_index[u] = len(self.T)
-            self.T.append(ValueTransition(tree_node=u, value=1))
-        while self.tree.next():
-            self.update_tree()
-            for site in self.tree.sites():
+        self.initialise(value=1)
+        while self.tree.prev():
+            self.update_tree(direction=tskit.REVERSE)
+            for site in reversed(list(self.tree.sites())):
                 self.process_site(site, h[site.id])
         return self.output
 
 
-class BackwardAlgorithm(LsHmmAlgorithm):
+class MirroredBackwardAlgorithm(LsHmmAlgorithm):
     """Runs the Li and Stephens backward algorithm."""
 
     def __init__(
@@ -690,13 +685,10 @@ def process_site(self, site, haplotype_state):
                 st.value = round(st.value, self.precision)
 
     def run(self, h):
-        self.tree.clear()
-        for u in self.ts.samples():
-            self.T_index[u] = len(self.T)
-            self.T.append(ValueTransition(tree_node=u, value=1))
-        while self.tree.prev():
-            self.update_tree(direction=tskit.REVERSE)
-            for site in reversed(list(self.tree.sites())):
+        self.initialise(value=1)
+        while self.tree.next():
+            self.update_tree()
+            for site in self.tree.sites():
                 self.process_site(site, h[site.id])
         return self.output