According to Tunisian olive museum (Zaitounah Museum) the olive tree came to Palestine from Armenia after which it spread to the Mediterranean and North Africa. It is explained in the following news article from 2004.
Zaitounah Museum new addition to Tunisian heritage
The Zaitounah (olive) Museum which was opened recently in the eastern city of Sousse is considered a new addition to the Tunisian cultural heritage due to the pivotal role the olive tree plays in the lives of Tunisians and the population of the Medieterranean region. […] Gadira said that it also aims at preserving the heritage of this “blessed” tree for future generations, noting that this tree remained steadfast before the various civilizations which ruled Tunisia over some 3000 years like the Berbers, Phoenicians, Byzantines, Arabs, Spaniards and Turks. He said that the most ancient documented sources available report that the olive tree was brought into Palestine from Armenia 4000 BC and then taken by the Phoenicians to Greece and later to North Africa, particularly Tunisia. Since the ancient times, olive oil was the choice of the elite and notables in Tunisia and other countries of the region. It was used in religious rituals, as a massage oil, producing perfumes and for other medical purposes. According to the latest figures, there are an estimated 55 million olive trees stretching from the country’s north to south and covering an area of nearly 1.6 million hectares or 30 percent of Tunisia’s farmlands. Tunisia is the world’s second largest producer and exporter of olive oil after EU states, mainly Italy and Spain. Article originally published by Kuwait News Agency (KUNA) 12-Jul-04