What’s the Origin of the Apricot Forest clinic’s
Japanese name, 香杏舎（Kokyosha）?

A legend of the Apricot Forest:

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In ancient China there lived a skilled doctor, named Toho（董奉）. Each time he cured someone, he ordered his patient to plant one apricot tree. Toho’s skill was so excellent that many apricot trees were planted.  After a decade, these planted trees became an apricot forest.
Thanks to this legend, “apricot” came to mean “skilled doctor” in China. When Dr. Higasa has opened his clinic, his teacher of Chinese ｍedicine, Dr. Yamamoto gave it the name Kokyosha.  He hoped it would remind both the staff of the clinic and its patience of the healing fragrance of apricot.
The above painting is a scene of Toho walking around apricot wood and.  It was drawn by Mr. Uemori, a professional painter who lives in France and paints landscapes of the Loire region. His touch is gentle and filled with classic Impressionism. Dr. Higasa loves his work and asked him to create this painting about the legend of the apricot forest.

More detail about this picture.