* 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.