* Iran's two neighbours face Taliban insurgencies

* Three presidents expected to also discuss drugs fight

By Hossein Jaseb

TEHRAN, May 24 (Reuters) - The presidents of Iran, Afghanistan and Pakistan began talks in Tehran on Sunday on ways to combat terrorism, drug trafficking and other regional security problems.

The one-day summit, part of efforts to boost cooperation between the three neighbours, coincides with an offensive launched by Pakistani security forces this month to stop the spread of a Taliban insurgency in the country's northwest.

Afghanistan, where violence has grown dramatically in the past two years despite the deployment of more U.S. and other foreign troops, is also battling Taliban militants.

Iranian President Mahmoud Ahmadinejad hosted his Pakistani and Afghan counterparts, Asif Ali Zardari and Hamid Karzai, at the meeting in northern Tehran.

"The presidents of the three countries will discuss security issues and the reconstruction of Afghanistan," Iranian state broadcaster IRIB said. Iran's Press TV said they would also talk about the opium trade, which helps fund the Taliban.

In Kabul last month, their foreign ministers agreed to meet once a month as part of efforts to fight terrorism and stabilise Afghanistan.

The United States has said it wants to increase its engagement with both Iran and Pakistan as part of a more regional approach to tackling the growing strength of Taliban militants across the south and east of Afghanistan.

MISTRUST

Despite three decades of mutual mistrust, analysts say Iran and the United States share an interest in securing regional stability. Iran says Washington is failing in Afghanistan, but that Tehran is ready to help its eastern neighbour.

At a U.N. meeting in The Hague in March, Iran offered to help Afghanistan combat the drugs trade, in a gesture that U.S. Secretary of State Hillary Clinton called promising.

The United States is pouring thousands of troops into Afghanistan this year to try to reverse gains by a resurgent Taliban, particularly in its southern heartland.

In Pakistan on Saturday, the military said street fighting erupted in the main town of the Swat valley as security forces mounted a new phase of their offensive against the militants.

The United States, which sees Pakistan as vital to its plan to defeat al Qaeda and bring stability in Afghanistan, has applauded Pakistani resolve to fight what some U.S. leaders have called an "existential threat" to the country.

The United Nations launched an appeal on Friday for $543 million for more than 2 million people displaced by fighting in northwest Pakistan, where officials said villagers were turning against the Taliban. (Writing by Fredrik Dahl; editing by Andrew Roche)

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