Bairstow made a brief statement after the match, saying there was no malice in his action and no animosity between the players.
Bancroft agreed there was no malice, but said he had been surprised by Bairstow's method of greeting at their first meeting.
The incident only came to light when some Australian fielders made a reference to the head butt — picked up on a stump microphone — while Bairstow was batting on Sunday, not long before he was dismissed to a regulation catch.
The ECB said Bairstow wouldn't be punished for the incident, even in the wake of a separate episode in September outside a club in Bristol that has resulted in allrounder Ben Stokes missing the start of the Ashes tour.
England coach Trevor Bayliss said team policies on curfews would be reviewed because he didn't want his players giving the Australians any opportunities to distract them.
Bayliss said the incident was "blown out of proportion" and was a case of "a couple of young guys out on a night out and some silly things happened."
"As I said it gives the opposition team ammunition to put pressure on as well," he said.The Australians did exactly that late in the match to extend their unbeaten run at the Gabba to 29 years.The unbroken stand between Warner (87) and Bancroft capped the win, and was the biggest opening partnership in a successful fourth-innings run chase in test cricket, according to Cricinfo.
It beat the mark of 172 set by Australians Archie Jackson and Bill Ponsford against the West Indies in 1930.
Smith's unbeaten 141 from 326 deliveries in the first innings earned him the player of the match award. The Australian skipper helped his team recover from 76-4 to reach 328 in reply to England's 302."It meant a lot. I really had to dig deep. It was tough, my slowest (century), but one of my best," Smith said.
"Cameron looked very good, had good plans, played very straight. The partnership with David was magnificent."The bowlers set up the easy chase with Nathan Lyon, Mitchell Starc and Josh Hazlewood taking three wickets apiece as England was bowled out for 195 on Sunday.
The second test — the first day-night test contested in an Ashes series — starts Saturday in Adelaide.
England holds the Ashes after a 3-2 win at home in 2015, but was swept 5-0 on its last tour to Australia in 2013-14
.After twice wiping out England's lower order in the match, and restricting the top order, the Australians are growing in confidence.England captain Joe Root said his team had outplayed Australia in the early stages, but hadn't taken their opportunities — particularly when Australia was in trouble in the first innings.
"Steve Smith played exceptionally well on that surface and took the game away from us," Root said. "For three days, we played some excellent cricket. Unfortunately, when we got into good positions, we didn't quite capitalize — had we done that, it would have been a very different scoreboard."There are things we need to address and learn from this game."
