There are, in my opinion, three answers to the question posed above. The first is a more generalized answer that could be applied to the importance of gardens in any continent. That is, subsistence. Gardens allowed peasants, craftsmen and merchants to subsidize their menial income with home grown food.
Source credit: http://www.senior-gardening.com/blogs/oldguy-blog.html
Of course, this answer is incomplete because all gardens are not planted specifically for subsistence, which brings us to the essence of the question at hand: why are gardens an intrinsic part of European culture.
The intrinsic nature of the garden in Europe is epitomized by the dissemination and evolution of religion. To take one example, the Roman gardens often contained elements such as statues of gods and goddesses as well as aediculas (small shrines to the gods or religious figures). As the Romans converted to Christianity they carried with them the sacred characteristics of the garden, i.e. the garden as a representation of the Garden in Eden. Even when the fall of the Roman empire transitioned Europe into the Medieval ages, the garden remained intricately tied to religion. Monasteries created utilitarian herb and vegetable gardens that provided food for the monks. As the church began to amass large sums of money the garden and the meaning and function therein changed. Monasteries started to build cloisters, which were gardens dedicated to the studying, teaching of religious themes and often featured religious themes: the tree of life, the four rivers of life, etc.
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Religion is interwoven throughout the fabric of European history and a natural counterpart to religion was the garden, hence the gardens intrinsic place in European culture.
However, the church was not the only institution that solidified the garden as an intrinsic aspect of European culture. Royalty and Nobility make up the final piece of this puzzle. From early ancient Europe, gardens have been associated with wealth and power: the palace of Versailles, the garden at the Chateau Villandry and Herrenhausen Gardens in Hannover, to name just a few.
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Due to the Europe’s history of turbulent royal regimes and the necessity of establishing power and wealth in the threat of domination, the garden was permanently sewn into the fabric of European culture.
Subsistence, religion and royalty are the three prongs to the intrinsic nature of the garden in Europe.
Check out this wikipedia article for more information on the history of gardens: http://en.wikipedia.org/wiki/History_of_gardening#European_gardens