diff --git a/cheuph/element_supply.py b/cheuph/element_supply.py
index 5f170e3..885a969 100644
--- a/cheuph/element_supply.py
+++ b/cheuph/element_supply.py
@@ -1,5 +1,5 @@
 import abc
-from typing import Dict, List, Optional
+from typing import Dict, List, Optional, Set
 
 from .element import Element, Id
 from .exceptions import TreeException
@@ -170,7 +170,7 @@ class MemoryElementSupply(ElementSupply):
 
     def __init__(self) -> None:
         self._elements: Dict[Id, Element] = {}
-        self._children: Dict[Id, List[Id]] = {}
+        self._children: Dict[Optional[Id], Set[Id]] = {None: set()}
 
     def add(self, element: Element) -> None:
         """
@@ -181,10 +181,8 @@ class MemoryElementSupply(ElementSupply):
             self.remove(element.id)
 
         self._elements[element.id] = element
-        self._children[element.id] = []
-
-        if element.parent_id is not None:
-            self._children[element.parent_id].append(element.id)
+        self._children[element.id] = set()
+        self._children[element.parent_id].add(element.id)
 
     def remove(self, element_id: Id) -> None:
         """
@@ -197,9 +195,7 @@ class MemoryElementSupply(ElementSupply):
 
             self._elements.pop(element_id)
             self._children.pop(element_id)
-
-            if element.parent_id is not None:
-                self._children[element.parent_id].remove(element.id)
+            self._children[element.parent_id].remove(element.id)
 
     def get(self, element_id: Id) -> Element:
         result = self._elements.get(element_id)
@@ -210,13 +206,9 @@ class MemoryElementSupply(ElementSupply):
         return result
 
     def get_children_ids(self, element_id: Optional[Id]) -> List[Id]:
-        result: Optional[List[Id]]
-        if element_id is None:
-            result = list(element.id for element in self._elements.values())
-        else:
-            result = self._children.get(element_id)
+        result = self._children.get(element_id)
 
         if result is None:
             raise TreeException(f"Element with id {element_id!r} could not be found")
 
-        return result
+        return list(sorted(result))