The FTSE in London closed ahead by 0.8% and the DAX in Germany added 0.3%.
The benchmark 10-year Treasury fell 22/32, raising the yield to 1.665%. The dollar was off by 0.35%, according to the dollar index.
On the economic front, the National Federation of Independent Business said its small-business optimism index dipped 0.1 points in May to 94.4. The reading was above a consensus view of 94.5, according to Ian Shepherdson, chief U.S. economist at High Frequency Economics.
"This is a great relief to us; the sales and economic expectations components of the survey are sensitive to stock prices so we feared a significant decline," he wrote. "The danger is not over yet, given the variability of the links between movements in stock prices and the survey, but so far the damage is slight ... Overall, small firm recovery continues, fitfully."Also, the Bureau of Labor Statistics reported that U.S. import prices fell 1% in May, as expected, after no change the previous month. Lower fuel and non-fuel prices contributed to the May decrease in overall import prices. Export prices also declined in May, falling 0.4% after a 0.4% increase in April. "Overall, a modest read on inflation pressure -- keeping with the trend as energy prices continue to decline," said Ian Lyngen senior bond strategist at CRT Capital. The Treasury Department's budget report found the May federal budget ran a deficit at $124.6 billion, which was higher against a $57.6 billion deficit during the same period a year ago. Hong Kong's Hang Seng index settled down 0.4% on Tuesday and the Nikkei in Japan shed 1%. July crude oil futures added 78 cents to settle at $83.48 a barrel. August gold settled up $17 at $1,613.80 an ounce. In corporate news, FactSet Research (FDS) reported third-quarter net income Tuesday of $48 million, or $1.05 a share, up from year-earlier earnings of $43.3 million, or 92 cents a share. On average, analysts anticipated third-quarter earnings of $1.04 a share. The company anticipates fourth-quarter non-GAAP earnings per share of between $1.15 and $1.17. Analysts, on average, forecast earnings of $1.08 a share. Shares tumbled more than 12%.