And his son was born in 1701.
I have looked at this before, but don't have edit access to the profile, and never did anything about it.
As I recall, R' Ephraim died c.1720, rather than was born in 1720. He died shortly before his daughter married the Besht. So she was born earlier than 1717. Probably more like 1700.
There have also been questions raised previously why R' Ephraim's son's matzeiva does not say he was a Cohen. See http://www.geni.com/discussions/78684
על פי ספר כתר כהונה (תודות רבנו הש"ך) לשבתי הכהן כץ היו 2 בנים ו-2 בנות. בעמוד 27 לספר נאמר כי שמות 2 חתניו היו מנחם (התחתן עם אסתר) ואהרון (שהתחתן עם טויבה).שמות בניו היו: הבכור משה והקטן מאיר.
בכל מקרה לא היה לו בן או חתן בשם אפרים ובכל הספר לא מוזכר השם הרב אפרים מברודי, כך שאפשר למחוק את השם הזה מתולדות הרב ש"ך.
And what is the source of the diagram at http://en.wikipedia.org/wiki/Baal_Shem_Tov_family_tree which shows הרב אפרים מברודי as the son of שבתי הכהן כץ?
The diagram from Wikipedia that shows R' Ephaim as the son of the Shach my indeed be misleading. Shimon's theory that the Shach was R' Ephaim's wife's father is interesting.
Wikipedia took the diagram from:
Brim, M. S., "כתר שם טוב: אילן היוחסין ... של צאצאי ... הבעש"ט" "Keter Shem Tov: A family tree of the descendants of the Baal Shem Tov (I)" (Family tree of the descendants of the Baal Shem Tov), in Heichal Habesht 23, pp. 164-182
Most other sources mention that R' Ephraim is "possibly" a son of the Shach. eg.
Menachem, I changed Rabbi Efrayim of Brody birthday to circa 1675 to be in line with the information regarding his his R' Gershon whose Yahrzeit is Adar 25th.
25 Adar - Rav Gershon Kitover, brother-in-law of the Baal Shem Tov (1696-1761).
His father, Efrayim, was a Rav and Av Beis Din in one of the four batei din in Brody, Poland.
Assuming there is a connection between the Shach and Rw Ephraim we all aggree that it is very unlikely they were father and son since Kehuna is a most important and revered quality that cannot be "invented" or "make dissapear"
Then the connection is probably through a female descendant
The Besht wife was born around 1700and according to the GENI tree is suppossed to be the Shach grandaughter But the are 80 years apart much more than 2 generations.
My theory is that somehow they are connected but I really do not know how
Is it possible that R' Gershon and his father were ostracized for following the Baal Shem Tov----his father R' Ephraim who made the match had died, and so both were left of sefer approbations etc, and thus the complication in finding their ancestors, except for the indirect hint of the connection to the SHaCH's family?
R'Gershon Kitover was born in Dubno, and the Meir ben Moses Ashenazi, the father of the SHaCH died in Dubno. So, there very likely could have been a non-Cohen connection.