  The density of a subgroupoid with respect to a free groupoid is defined as
the asymptotic ratio of their growths. This notion can be interpreted as a
generalisation of the index's inverse for groups or as the probability of an
element belonging to a subgroupoid. This more combinatorial strategy shows a
richer picture of free groupoids than the bare algebraic perspective. We study
the growth and density of several subgroupoids of a free groupoid. In addition,
some aspects of enumeration related to the Motzkin paths are shown.
