diff -up src/sage/misc/binary_tree.pyx.orig src/sage/misc/binary_tree.pyx
--- src/sage/misc/binary_tree.pyx.orig	2020-10-24 11:37:59.000000000 -0600
+++ src/sage/misc/binary_tree.pyx	2020-10-30 13:27:44.173427025 -0600
@@ -12,6 +12,8 @@ from cysignals.memory cimport sig_malloc
 
 from cpython.ref cimport PyObject, Py_INCREF, Py_XDECREF
 
+from sage.misc.prandom import randint
+
 cdef binary_tree_node *BinaryTreeNode(int key, object value):
     cdef binary_tree_node *t
     t = <binary_tree_node *>sig_malloc(sizeof(binary_tree_node))
@@ -121,10 +123,7 @@ cdef binary_tree_node *binary_tree_right
 cdef binary_tree_node *binary_tree_head_excise(binary_tree_node *self):
     cdef binary_tree_node *cur
     cdef int right
-    # We have a pointer we're about to free.  Chances are, we'll never
-    # see this pointer again.  Thus, its least significant bit is
-    # "random" enough to resist bias.
-    right = (<int>self)&1
+    right = randint(0, 1)
     if self.right == NULL:
         return self.left
     if self.left == NULL:
@@ -501,7 +500,6 @@ class Test:
 
             sage: sage.misc.binary_tree.Test().random()
         """
-        from sage.misc.prandom import randint
         t = BinaryTree()
         for i in xrange(cycles):
             r = randint(0,8)