"tired, exhausted," 1870, American English, perhaps from earlier sense of "lost in the woods" (1856), from bush (n.).
Entries linking to bushed
"many-stemmed woody plant," from Old English bysc (found in place names), from West Germanic *busk "bush, thicket" (source also of Old Saxon and Old High German busc, Dutch bosch, bos, German Busch). Influenced by or combined with Old French (busche "firewood") and Medieval Latin busca (source also of Italian bosco, Spanish bosque, French bois), both of which probably are from Germanic (compare Boise).
In the British American colonies, applied from 1650s to the uncleared districts. In South Africa, "country," as opposed to town (1780); probably from Dutch bosch in the same sense. As "branch of a tree hung out as a tavern-sign," 1530s; hence the proverb "good wine needs no bush." Meaning "pubic hair" (especially of a woman) is from 1745. To beat the bushes (mid-15c.) is a way to rouse birds so that they fly into the net which others are holding, which originally was the same thing as beating around the bush (see beat (v.)).