THIRTEEN Tibetan nuns and monks were arrested for taking part in pro-independence demonstrations in the past two months, according to the London-based Tibet Information Network. The three demonstrations, though short-lived and small in scale, took place in Lhasa, the capital of Tibet, or in the region. On March 21, three nuns gathered in Barkor Square, Lhasa, and began to chant slogans calling for ''freedom for Tibet'' and for the long life of the Dalai Lama. Some called for ''religious freedom'' and asked the Chinese ''to leave and go back to China''. The protesting nuns were taken away by police and detained in a prison to the east of Lhasa. A Tibetan source said the nuns seemed about 18 years old. They came from a nunnery in the rural county of north Lhasa where more than 30 women were serving long prison sentences for taking part in protests. A second incident took place on March 27 when two young monks staged a brief demonstration in the pilgrimage circuit surrounding the main temple in Lhasa. The monks were arrested as soon as the security forces discovered the monks were carrying a Tibetan flag, which was forbidden there. A more serious protest was reported after eight nuns were arrested in Kyimshi, one of several villages to the south of Lhasa, last February. It was likely that the nuns came from a nunnery in Kyimshi. In 1993, 11 of the 42 pro-independence protests reported in Tibet took place outside Lhasa, and some Tibetan activists in Lhasa claimed that unrest was growing in the countryside. ''The rural population are showing themselves to be every bit more alert and lucid, unlike in the past few years, and they no longer swallow so easily any of the soft and gentle politics played on them by the state,'' said one Tibetan activist in Lhasa last January.