The house of Hayashi was one of the Four houses.
The heads of the Hayashi Go School (林家) (1612-1884) were:
Hayashi in Japanese means woods. It is said that when the Shogun offered a surname to this school, he suggested Mori 森, or forest. The recipient modestly declined and chose instead the more humble Hayashi.
John F. I think a better way of describing Shuei's translation back to the Honinbo family could be found, as the Hayashi family did not then cease. A collateral branch came down through Yubi - Hayashi Sano was the 16th generation, for example, which probably makes the Hayashi family the first to have a notable female player.