Project: During the study of planar trees J.-L. Loday and M.O. Ronco introduced a type of algebras, called trialgebras, which are vector spaces endowed with three binary associative operations satisfying eight axioms. A trialgebra is just a linear analog of a trioid. J.-L. Loday and  M.O. Ronco constructed a free trioid of rank 1. Later, free trioids of an arbitrary rank and some relatively free trioids were given. In this project our attention will be aimed to studying free trioids. The main aims are to characterize the structure of free trioids, some congruences on them and describe properties of underlying semigroups of free trioids. The obtained  results  can be applied to trialgebra theory.