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The pair racked up a 210-run partnership as England reached 267 for two at stumps at Wellington's Basin Reserve, with Trott unbeaten on 121 and Kevin Pietersen 18 not out after Compton notched back-to-back Test centuries.
England punished New Zealand captain Brendon McCullum for sending them in to bat after winning the toss on a wicket that defied expectations and proved flat and true as cloud cover gave way to blue sky.
It provided perfect conditions for the tourists, who contented themselves with steady progress, determined not to repeat the mistakes of the drawn first Test, when they were bowled out for 167 in the first innings.
Alastair Cook's dismissal in the first hour was the only blemish, with the England captain misjudging an inswinging delivery from Neil Wagner and scooping the ball to Peter Fulton at mid-on to depart for 17.
But his dismissal merely cleared the way for Compton and Trott to take charge, batting together for 63 overs to demoralise New Zealand's bowlers, who toiled relentlessly for cant reward on the docile pitch.
The consistent Trott batted through most of the day to bring up his ninth Test century in the final session with a pull shot to the boundary, his 14th four of the innings.
Compton followed with his own ton soon after, dispatching Wagner to the ropes with a classic cover drive for his second century in successive innings.
He was out for 100 two overs later when an attempt to drive Bruce Martin caught an edge and carried to Ross Taylor in the slips.
But the assured innings is likely to consolidate the 29-year-old's spot at the top of the order, which had been under pressure from Joe Root.
There were flashes of aggression when Compton and Trott resumed after lunch with England at 75-1. They clubbed 40 in a six-over spell before spinner Martin was brought on to stem the flow of runs.
The next 40 runs took more than 20 overs, with England happy to blunt the New Zealand attack and build towards a big total. Pietersen curbed his big-hitting instincts when he replaced Compton, making his way to 18 off 48 balls after disappointing returns of 0 and 12 in the first Test. England, ranked second in Test cricket, are favourites in the three-Test series.
New Zealand, ranked eighth, have not won a series against England since 1999, with their sole series victory on home soil back in 1984.
However, McCullum will be disappointed at the way his side lost momentum after dominating much of the first Test.