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As is evident from 26:176 ff., the "dwellers of the wooded dales (al-aykah)" were the people of Madyan, who refused to pay heed to their prophet, Shu'ayb, and were thereupon destroyed, apparently by an earthquake and/or a volcanic eruption (cf. {7:85-93} and {11:84-95}).
"Companions of the Wood": As-hab ul Aikati. Perhaps Aika is after all a proper noun, the name of a town or tract. Who were the Companions of the Aika? They are mentioned four times in the Qur-an, viz., here, and in xxvi. 176-191; xxxviii. 13; and 1. 14. The only passage in which any details are given is xxvi. 176-191. There we are told that their Prophet was Shu'aib, and other details given correspond to those of the Madyan, to whom Shu'aib was sent as Prophet;, see vii. 85-93. In my notes to that passage I have discussed the question of Shu'aib and the Madyan people. It is reasonable to suppose that the Companions of the Wood were either the same as the Madyan, or a Group among them or in their neighbourhood.