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Investors bet that Trump would not push through with extreme protectionist policies, and stocks in big European carmakers were the main beneficiaries of the relief rally.
"Stocks bounce despite trade war concerns," Craig Erlam, senior market analyst at OANDA said, while Jasper Lawler at London Capital Group said that "risk-on trading" was making a return.
But Wall Street turned lower in late New York morning trade, having given up opening gains, helping push European markets off the day's best levels.
"US stocks are struggling to extend yesterday's solid gains as global trade uncertainty lingers," analysts at the Charles Schwab brokerage said.
Asia had surged earlier as shock over the controversial tariffs announcement gave way to hope that Trump may not have the full backing of his Republican party on this move.
According to Lawler, the impact of Trump's announcement may have been dampened "thanks to relatively little interest by China in retaliating against the US".
Markets were also buoyed by an announcement from Seoul that the leaders of North and South Korea will hold a historic summit in the Demilitarised Zone next month after Pyongyang expressed willingness to give up its nuclear weapons.
"Risk appetite has improved on headlines North Korea has expressed a willingness to talk about de-nuclearisation, according to South Korea," analyst Fawad Razaqzada of Forex said.
Lukman Otunuga, research analyst at FXTM, described as "remarkable and somewhat alarming how the risk-off sentiment last Friday was swiftly replaced with a risk-on mood today".
"European stocks were elevated by the renewed appetite for risk," he added.
Republican Trump had sparked fears of a global trade war late last week when he unveiled plans to slap levies on imports of steel and aluminium, sending share prices tumbling.
European automakers revved higher, after losing speed on worries that a trade war could hurt valuable exports into the United States.
In Frankfurt, carmaker Volkswagen jumped by more than three per cent, with Daimler and BMW also posting gains.
"A strategic target - and standing to lose the most from any possible trade war - the German auto sector is rebounding sharply, helping lift the DAX higher," noted Interactive Investor analyst Rebecca O'Keeffe.
In Paris, Renault and Peugeot were also higher.
And Milan's benchmark FTSE MIB shares index wiped out the previous day's losses seen after Italy's inconclusive election outcome, and even added hefty gains, outperforming its European peers.
London - FTSE 100: UP 0.4 per cent at 7,146.75 points (close)
Frankfurt - DAX 30: UP 0.2 per cent at 12,113.87 (close)
Paris - CAC 40: UP 0.1 per cent at 5,170.23 (close)
Milan - FTSE MIB: UP 1.8 per cent at 22,196.99
EURO STOXX 50: UP 0.1 per cent at 3,357.86
New York - Dow: DOWN 0.5 per cent at 24,741.42
Tokyo - Nikkei 225: UP 1.8 per cent at 21,417.76 (close)
Hong Kong - Hang Seng: UP 2.1 per cent at 30,510.73 (close)
Shanghai - Composite: UP 1.0 per cent at 3,289.64 (close)
Euro/dollar: UP at US$1.2410 from US$1.2336 at 2200 GMT on Monday
Pound/dollar: UP at US$1.3886 from US$1.3849
Dollar/yen: DOWN at 105.93 yen from 106.20 yen
Oil - Brent North Sea: DOWN 10 cents at US$65.44 per barrel
Oil - West Texas Intermediate: DOWN 10 cents at US$62.47 per barrel