brushyoudaoicibaDictYouDict[brush 词源字典]
brush: [14] It is not clear whether brush for sweeping and brush as in brushwood are the same word, although both appeared in the language at about the same time, from a French source. Brush ‘broken branches’ comes from brousse, the Anglo-Norman version of Old French broce, which goes back to an unrecorded Vulgar Latin *bruscia. Brush for sweeping, on the other hand, comes from Old French broisse or brosse.

It is tempting to conclude that this is a variant of Old French broce, particularly in view of the plausible semantic link that brushwood (cut twigs, etc) bundled together and tied to a handle makes a serviceable brush (that is how broom came to mean ‘brush’). The verb brush ‘move fast or heedlessly’ comes from Old French brosser ‘dash through undergrowth’, a derivative of broce; its frequent modern connotation of ‘touching in passing’ comes from the other brush.
[brush etymology, brush origin, 英语词源]
brush (v.1)youdaoicibaDictYouDict
late 15c., "to clean or rub (clothing) with a brush," also (mid-15c.) "to beat with a brush," from brush (n.1). Related: Brushed; brushing. To brush off someone or something, "rebuff, dismiss," is from 1941.
brush (n.1)youdaoicibaDictYouDict
"dust-sweeper, a brush for sweeping," late 14c., also, c. 1400, "brushwood, brushes;" from Old French broisse (Modern French brosse) "a brush" (13c.), perhaps from Vulgar Latin *bruscia "a bunch of new shoots" (used to sweep away dust), perhaps from Proto-Germanic *bruskaz "underbrush."
brush (n.2)youdaoicibaDictYouDict
"shrubbery," early 14c., from Anglo-French bruce "brushwood," Old North French broche, Old French broce "bush, thicket, undergrowth" (12c., Modern French brosse), from Gallo-Roman *brocia, perhaps from *brucus "heather," or possibly from the same source as brush (n.1).
brush (v.2)youdaoicibaDictYouDict
"move briskly" especially past or against something or someone, 1670s, from earlier sense (c. 1400) "to hasten, rush," probably from brush (n.2), on the notion of a horse, etc., passing through dense undergrowth (compare Old French brosser "travel (through woods)," and Middle English noun brush "charge, onslaught, encounter," mid-14c.), but brush (n.1) probably has contributed something to it as well. Related: Brushed; brushing.