sagemathgh-39519: Provide an iterative version on some functions on trees
    
This pull request provide iterative methods on AbstractTrees which work
both on OrderedTree and BinaryTree.
The problem was that some functions like node_number reach the recursion
limit on huge instances.
```
sage: T = OrderedTree([])
sage: for _ in range(10000):
....:     T = OrderedTree([])
sage: T.node_number()
RecursionError
```

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.
    
URL: sagemath#39519
Reported by: Oscfon
Reviewer(s): Frédéric Chapoton, Martin Rubey, Oscfon

Author: Release Manager

src/sage/combinat/abstract_tree.py

@@ -276,8 +276,7 @@ def action(x):
         while stack:
             node = stack.pop()
             action(node)
-            for i in range(len(node)):
-                subtree = node[-i - 1]
+            for subtree in reversed(node):
                 if not subtree.is_empty():
                     stack.append(subtree)
 
@@ -639,15 +638,122 @@ def action(x):
                 # subtrees, and should not be exploded again, but instead
                 # should be manipulated and removed from the stack.
                 stack.append(None)
-                for i in range(len(node)):
-                    subtree = node[-i - 1]
+                for subtree in reversed(node):
                     if not subtree.is_empty():
                         stack.append(subtree)
             else:
                 stack.pop()
                 node = stack.pop()
                 action(node)
 
+    def contour_traversal(self, first_action=None, middle_action=None, final_action=None, leaf_action=None):
+        r"""
+        Run the counterclockwise countour traversal algorithm (iterative
+        implementation) and subject every node encountered
+        to some procedure ``first_action``, ``middle_action`` or ``final_action`` each time it reaches it.
+
+        ALGORITHM:
+
+        - if the root is a leaf, apply `leaf_action`
+        - else
+            -  apply `first_action` to the root
+            - iteratively apply `middle_action` to the root and traverse each subtree
+              from the leftmost one to the rightmost one
+            - apply `final_action` to the root
+
+        INPUT:
+
+        - ``first_action`` -- (optional) a function which takes a node as
+          input, and does something the first time it is reached during exploration
+
+        - ``middle_action`` -- (optional) a function which takes a node as
+          input, and does something each time it explore one of its children
+
+        - ``final_action`` -- (optional) a function which takes a node as
+          input, and does something the last time it is reached during exploration
+
+        - ``leaf_action`` -- (optional) a function which takes a leaf as
+          input, and does something when it is reached during exploration.
+
+        OUTPUT:
+
+        ``None``. (This is *not* an iterator.)
+
+        TESTS::
+
+            sage: l = []
+            sage: t = OrderedTree([[],[[],[],]]).canonical_labelling()
+            sage: t
+            1[2[], 3[4[], 5[]]]
+            sage: t.contour_traversal(lambda node: (l.append(node.label()),l.append('a')),
+            ....:   lambda node: (l.append(node.label()),l.append('b')),
+            ....:   lambda node: (l.append(node.label()),l.append('c')),
+            ....:   lambda node: (l.append(node.label())))
+            sage: l
+            [1, 'a', 1, 'b', 2, 1, 'b', 3, 'a', 3, 'b', 4, 3, 'b', 5, 3, 'c', 1, 'c']
+
+            sage: l = []
+            sage: b = BinaryTree([[None,[]],[[[],[]],[]]]).canonical_labelling()
+            sage: b
+            3[1[., 2[., .]], 7[5[4[., .], 6[., .]], 8[., .]]]
+            sage: b.contour_traversal(lambda node: l.append(node.label()),
+            ....:   lambda node: l.append(node.label()),
+            ....:   lambda node: l.append(node.label()),
+            ....:   None)
+            sage: l
+            [3, 3, 1, 1, 1, 2, 2, 2, 2, 1, 3, 7, 7, 5, 5, 4, 4, 4, 4, 5, 6, 6, 6, 6, 5, 7, 8, 8, 8, 8, 7, 3]
+
+        The following test checks that things do not go wrong if some among
+        the descendants of the tree are equal or even identical::
+
+            sage: u = BinaryTree(None)
+            sage: v = BinaryTree([u, u])
+            sage: w = BinaryTree([v, v])
+            sage: t = BinaryTree([w, w])
+            sage: t.node_number()
+            7
+            sage: l = []
+            sage: t.contour_traversal(first_action = lambda node: l.append(0))
+            sage: len(l)
+            7
+        """
+        if first_action is None:
+            def first_action(x):
+                return
+        if middle_action is None:
+            def middle_action(x):
+                return
+        if final_action is None:
+            def final_action(x):
+                return
+        if leaf_action is None:
+            def leaf_action(x):
+                return
+        stack = []
+        stack.append(self)
+        corners = [0, 0]
+        while stack:
+            node = stack.pop()
+            if not node:
+                leaf_action(node)
+                corners.pop()
+                corners[-1] += 1
+            elif not corners[-1]:
+                first_action(node)
+                middle_action(node)
+                stack.append(node)
+                stack.append(node[0])
+                corners.append(0)
+            elif corners[-1] == len(node):
+                final_action(node)
+                corners.pop()
+                corners[-1] += 1
+            else:
+                middle_action(node)
+                stack.append(node)
+                stack.append(node[corners[-1]])
+                corners.append(0)
+
     def breadth_first_order_traversal(self, action=None):
         r"""
         Run the breadth-first post-order traversal algorithm
@@ -823,12 +929,41 @@ def node_number_at_depth(self, depth):
             True
             sage: [T.node_number_at_depth(i) for i in range(3)]
             [0, 0, 0]
+
+        Check that we do not hit a recursion limit::
+
+            sage: T = OrderedTree([])
+            sage: for _ in range(9999):
+            ....:     T = OrderedTree([T])
+            sage: T.node_number_at_depth(2000)
+            1
         """
         if self.is_empty():
-            return Integer(0)
-        if depth == 0:
-            return Integer(1)
-        return sum(son.node_number_at_depth(depth - 1) for son in self)
+            return 0
+        m = 0
+
+        def fr_action(node):
+            nonlocal m, depths, depth
+            if depths[-1] == depth:
+                m += 1
+
+        def m_action(node):
+            nonlocal depths
+            depths.append(depths[-1] + 1)
+
+        def fn_action(node):
+            nonlocal depths
+            depths.pop()
+
+        def lf_action(node):
+            nonlocal m, depths, depth
+            if depths[-1] == depth:
+                m += 1
+            depths.pop()
+
+        depths = [0]
+        self.contour_traversal(fr_action, m_action, fn_action, lf_action)
+        return Integer(m)
 
     def paths_to_the_right(self, path):
         r"""
@@ -1052,11 +1187,25 @@ def node_number(self):
             2
             sage: BinaryTree([[None, [[], []]], None]).node_number()
             5
+
+        TESTS:
+
+        Check that we do not hit a recursion limit::
+
+            sage: T = OrderedTree([])
+            sage: for _ in range(9999):
+            ....:     T = OrderedTree([T])
+            sage: T.node_number()
+            10000
         """
-        if self.is_empty():
-            return Integer(0)
-        else:
-            return sum((i.node_number() for i in self), Integer(1))
+        count = 0
+
+        def incr(node):
+            nonlocal count
+            count += 1
+
+        self.iterative_pre_order_traversal(incr)
+        return Integer(count)
 
     def depth(self):
         """
@@ -1079,11 +1228,33 @@ def depth(self):
             0
             sage: BinaryTree([[],[[],[]]]).depth()
             3
+
+        TESTS:
+
+        Check that we do not hit a recursion limit::
+
+            sage: T = OrderedTree([])
+            sage: for _ in range(9999):
+            ....:     T = OrderedTree([T])
+            sage: T.depth()
+            10000
         """
-        if self:
-            return Integer(1 + max(i.depth() for i in self))
-        else:
-            return Integer(0 if self.is_empty() else 1)
+        if self.is_empty():
+            return 0
+        m = []
+
+        def action(node):
+            nonlocal m
+            if node.is_empty():
+                m.append(-1)
+            elif not bool(node):
+                m.append(0)
+            else:
+                mx = max(m.pop() for _ in node)
+                m.append(mx + 1)
+
+        self.contour_traversal(final_action=action, leaf_action=action)
+        return Integer(m[0] + 1)
 
     def _ascii_art_(self):
         r"""