Ki (木) vs. Hayashi (林) vs. Mori (森): Meaning in Japanese
木 (Ki), 林 (Hayashi), and 森 (Mori) are Kanji characters that we learn in elementary school. They are similar in shape, from which you could guess the meanings.
This may be off topic, but although Hayashi (林) and Mori (森) are common Japanese family names, I have never heard of the one named Ki (木).
Ki vs. Hayashi vs. Mori
The Japanese word usually read as Ki (pronunciation), 木, consists of a single 木 and means a tree. Written in 4 strokes, the Kanji is simple and easy to understand.
However, 林/Hayashi (pronunciation) and 森/Mori (pronunciation) are not, even though these are both often translated into English as forest.
Hayashi (林) vs. Mori (森)
Based on Kotobank, it seems Hayashi and Mori were once quite clearly distinguished;
People called a natural forest close to human settlements 森, a forest altered by human hand 林, and forests in the middle of nowhere 山/Yama (meaning mountain).
Those classifications are still partially alive, but the divisions are now getting unclear.
Nonetheless, as said that the etymology of 森 is 盛り (meaning heap or forming naturally), many of us tend to associate Mori with a grove with a figure like 森.
Meanwhile, Hayashi, derived from 生やす/planted, is reminiscent of the woods with uniform heights, like 林, as explained in this article on Shinrin-Ringyou.com.
(Reference Pages: Goo Japanese Dictionary 森, 林 )