In Esperanto you can construct many new words from a relatively low number of root-word. Example, with from arbo (tree) and aro (set) you can build arbaro (forest).

My question is three folded:

  • What is this phenomenon (building vocabulary from a relatively small number of root words) called?
  • Is there a metric or it in quantitative linguistic? What is its name?
  • Is there a table with a comparison for different languages, included non-natural ones?

Your question is about derivational morphology. There are mainly three strategies to create new words: 1. add affixes to a stem (as in your example), 2. combine stems (as in Chinese), 3. modify the vowels (as in Semitic). Purely agglutinative languages only apply strategy1. Indo-European languages tend to apply a mixture of all three strategies, for example in German. Latin (and Romance) mostly applies 1 and 2 (strategy3 is fossilized in Latin). Note that Esperanto also applies strategy 2, but not strategy 3, as far as I know. Semitic applies a mixture of strategy 1 and 3, as in maktub < ktb, but not strategy2. I'm not sure it's possible to make quantitative surveys of these phenomena. It's more about typological tendencies.

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On the second question, I think measures of morphological productivity are what you're looking for. If you search around, you should be able to find information on productivity measures easily, e.g. this overview:

Baayen, R. H. (2009). 43. Corpus linguistics in morphology: morphological productivity. Corpus linguistics: An international handbook, 900-919.

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