Did the Buddha form a monastery? According to Wikipedia's Buddhist monasticism article, the Buddha formed the first monastery. There is no source, 'though. To quote:

Buddhist Monasticism is one of the earliest surviving forms of organized monasticism in the history of religion.
The order of Buddhist monks and nuns was founded by Gautama Buddha during his lifetime over 2500 years ago.

Ok, so it doesn't specifically say 'monastery' here but still, an order of monks and nuns, some sort of precursor to a monastery at least.
According to History World's History of Monasticism:

Leading his followers into holiness a generation after Mahavira, Buddha also organizes them into communities.

Is this just &quot;according to tradition&quot; or is there historical evidence of him forming a monastery/order/community: such as archaeological findings or non-religious texts? If yes, is there's anything left of the original or has it all branched out?
The Buddha often taught in forest groves, the most famous being Isipatana (Deer park) where he delivered his first lecture (Dhammacakkappavattana Sutta), 



and Jetavana (Jeta Grove) (famously recorded in Buddhist epics as purchased by Anadapindika by paving the ground with gold) where there's still believed to be architectural remains of the Buddha's hut (Gandhakuti) from his stay there. Buddhist texts record that there were several imposing architectural structures constructed there, including an imposing arch, though almost none of it remains today.


  Some of the chief buildings attached to the Jetavana are mentioned in the books by special names, viz., Mahāgandhakuti, Kaverimandalamāla, Kosambakuti and Candanamāla. SNA.ii.403. Other buildings are also mentioned - e.g., the Ambalakotthaka-āsanasālā (J.ii.246). According to Tibetan sources the vihāra was built according to a plan sent by the devas of Tusita and contained sixty large halls and sixty small. The Dulva also gives details of the decorative scheme of the vihāra (Rockhill: op. cit.48 and n.2).


source: http://www.palikanon.com/english/pali_names/j/jetavana.htm



There's also Vulture's peak, frequently referred in the suttas as one of the Buddha's favoured places to retreat to.



Many of the other locations that are referenced in the suttas, or where ruins of buildings still exist feature prominently on buddhist pilgrimage routes.

In later periods very prominent Buddhist Viharas, or universities functioned for over a 1000 years at the same spots, namely at Taxila (present day Pakistan) and Nalanda (present day India). They housed at their peak thousands of monks and novices, living and training for years together.
Eminent personages such as Shantideva, the 8th-century Indian Buddhist monk and scholar emerged from these universities.