Preserving original doctype and declaration of an lxml.etree parsed xml

I'm using python's lxml and I'm trying to read an xml document, modify and write it back but the original doctype and xml declaration disappears. I'm wondering if there's an easy way of putting it back in whether through lxml or some other solution?

tl;dr

# adds declaration with version and encoding regardless of
# which attributes were present in the original declaration
# expects utf-8 encoding (encode/decode calls)
# depending on your needs you might want to improve that
from lxml import etree
from xml.dom.minidom import parseString
xml1 = '''\
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE root SYSTEM "example.dtd">
<root>...</root>
'''
xml2 = '''\
<root>...</root>
'''
def has_xml_declaration(xml):
    return parseString(xml).version
def process(xml):
    t = etree.fromstring(xml.encode()).getroottree()
    if has_xml_declaration(xml):
        print(etree.tostring(t, xml_declaration=True, encoding=t.docinfo.encoding).decode())
    else:
        print(etree.tostring(t).decode())
process(xml1)
process(xml2)

The following will include the DOCTYPE and the XML declaration:

from lxml import etree
from StringIO import StringIO

tree = etree.parse(StringIO('''<?xml version="1.0" encoding="iso-8859-1"?>
 <!DOCTYPE root SYSTEM "test" [ <!ENTITY tasty "eggs"> ]>
  <root>
   <a>&tasty;</a>
 </root>
'''))

docinfo = tree.docinfo
print etree.tostring(tree, xml_declaration=True, encoding=docinfo.encoding)

Note, tostring does not preserve the DOCTYPE if you create an Element (e.g. using fromstring), it only works when you process the XML using parse.

Update: as pointed out by J.F. Sebastian my assertion about fromstring is not true.

Here is some code to highlight the differences between Element and ElementTree serialization:

from lxml import etree
from StringIO import StringIO

xml_str = '''<?xml version="1.0" encoding="iso-8859-1"?>
 <!DOCTYPE root SYSTEM "test" [ <!ENTITY tasty "eggs"> ]>
  <root>
   <a>&tasty;</a>
 </root>
'''

# get the ElementTree using parse
parse_tree = etree.parse(StringIO(xml_str))
encoding = parse_tree.docinfo.encoding
result = etree.tostring(parse_tree, xml_declaration=True, encoding=encoding)
print "%s\nparse ElementTree:\n%s\n" % ('-'*20, result)

# get the ElementTree using fromstring
fromstring_tree = etree.fromstring(xml_str).getroottree()
encoding = fromstring_tree.docinfo.encoding
result = etree.tostring(fromstring_tree, xml_declaration=True, encoding=encoding)
print "%s\nfromstring ElementTree:\n%s\n" % ('-'*20, result)

# DOCTYPE is lost, and no access to encoding
fromstring_element = etree.fromstring(xml_str)
result = etree.tostring(fromstring_element, xml_declaration=True)
print "%s\nfromstring Element:\n%s\n" % ('-'*20, result)

and the output is:

--------------------
parse ElementTree:
<?xml version='1.0' encoding='iso-8859-1'?>
<!DOCTYPE root SYSTEM "test" [
<!ENTITY tasty "eggs">
]>
<root>
   <a>eggs</a>
 </root>

--------------------
fromstring ElementTree:
<?xml version='1.0' encoding='iso-8859-1'?>
<!DOCTYPE root SYSTEM "test" [
<!ENTITY tasty "eggs">
]>
<root>
   <a>eggs</a>
 </root>

--------------------
fromstring Element:
<?xml version='1.0' encoding='ASCII'?>
<root>
   <a>eggs</a>
 </root>