This is the 4018th most frequent Russian word.
лесной
"Forest" or "wooded" (related to a forest).
Here, 'лесной' modifies 'тропинке' to mean 'forest', indicating the location or type of the path.
Я гуляю по лесной тропинке.
I am walking along a forest path.
'Лесной' describes 'воздух', signifying that the air comes from or is characteristic of a forest.
Лесной воздух свежий и чистый.
The forest air is fresh and clean.
In this usage, 'лесной' relates to 'домик', specifying that the house is situated in or related to a forest.