If I understand this correctly you want the children of an armatures bones.
In that case the children are still the armature objects children. The bones are specified as obj.parent_bone given that obj.parent_type == “BONE”.
Edit: Changed obj.parent_type (mistake)
Here is a long version of how to handle it:
import bpy
List = []
ARM = bpy.data.objects["Armature"]
for BONE in ARM.data.bones: # search through bones
Bonechildren = []
for CHILD in ARM.children: # check if there are children of the armature that have the bone as parent_bone
if CHILD.parent_type == "BONE" and CHILD.parent_bone == BONE.name: # check for the name string, not bone data
Bonechildren.append(CHILD)
List.append([BONE, Bonechildren])
print(List) # [ [bone1, bone1_children] , [bone2, bone2_children] ... ]
for x in List: print(x)