TREASURIES-Shorter-dated yields mostly higher as crude prices climb; focus turns to labor data

* Crude prices climb, but gains capped by shipping optimism

* US labor market data due later this week

* Yield on 2-year note rises after four straight daily declines

By Chuck Mikolajczak

NEW YORK, June 29 (Reuters) - Shorter-dated U.S. Treasury yields were mostly higher on Monday, as crude prices rose following attacks between the U.S. and Iran over the weekend and ahead of a flurry of labor market data later this week. U.S. crude rose 1.34% to $70.15 a barrel and Brent rose to $72.77 per barrel, up 1.08% on the day as the attacks once again threatened a tenuous peace deal, although expectations of a continued recovery in energy shipping through the Strait of Hormuz kept gains in check. Iranian and U.S. technical teams working on the implementation of an interim peace deal are expected to meet in Doha in the coming days, a source told Reuters on Monday. Markets will see a string of data on the labor market this week, culminating with the release on Thursday of the Labor Department's monthly payrolls report for June. Yields have been declining in recent days, as expectations of easing inflation pressures have grown, offsetting what was seen as a hawkish Federal Reserve policy announcement and press conference by new Fed Chairman Kevin Warsh on June 17. "The labor market to me is really interesting, the data for this week, but the key, the focus now is more on the inflation side," said Jim Barnes, director of fixed income at Bryn Mawr Trust. "Because energy prices have materially come down, inflation expectations have notably come down. But the market now, especially after the Fed meeting, it's not good enough, now they have to start to see more concrete evidence that inflation's coming down."

BENCHMARK YIELDS EDGE HIGHER

The yield on the benchmark U.S. 10-year Treasury note edged up 0.4 basis point to 4.376% after having fallen for three straight weeks. Comments from several Fed officials indicated last week that they were still concerned about high inflation.

The yield on the 30-year bond shed 0.3 basis point to 4.862%. Markets are currently pricing in a 29.4% chance of a rate hike of at least 25 basis points at the Fed's July 28-29 meeting, according to CME Group's FedWatch tool, and a 61.9% chance at the September 15-16 meeting.

The two-year U.S. Treasury yield, which typically moves in step with interest rate expectations for the Fed, rose 1.9 basis points to 4.107% and was on track for its first daily gain after four straight declines.

A closely watched part of the U.S. Treasury yield curve measuring the gap between yields on two- and 10-year Treasury notes, seen as an indicator of economic expectations, was at a positive 26.7 basis points.

The breakeven rate on five-year U.S. Treasury Inflation-Protected Securities (TIPS) was last at 2.244% after closing at 2.223% on June 26.

The 10-year TIPS breakeven rate was last at 2.215%, indicating the market sees inflation averaging about 2.2% a year for the next decade. (Reporting by Chuck Mikolajczak; Editing by Paul Simao)