Lake City is located in Missaukee County, Michigan several counties north and east of Lake County.
Check "Michigan Births, 1867-1902" at FamilySearch.org: https://familysearch.org/search/collection/show#uri=https%3A...
Has digital images of original birth registers (state duplicate).
Also check the Missaukee Co. GenWeb site at: http://www.mifamilyhistory.org/missaukee/
I suspect that Bertha's 1884 birth was not recorded. Registration compliance before 1914 was low and many births were not recorded.