The universal property of the multitude of trees

A vital ingredient in the first author's definition of weak ω-category is his description, in terms of trees, of the free (strict) ω-category on a globular set. The induced monad on the category of globular sets shares many of the properties of the monoid monad (describable in terms of words) on the category of sets. Bénabou has shown how the simplicial category arises from the monoid monad. The present paper studies the object arising similarly from the ω-category monad.