  In a previous work, we gave a coalgebraic framework of directed graphs
equipped with weights (or probability vectors) in terms of (Markov)
L-coalgebras. They are K-vector spaces equipped with two co-operations,
\Delta_M, \tilde{\Delta}_M verifying,
  (\tilde{\Delta}_M \otimes id)\Delta_M =(id \otimes \Delta_M)\tilde{\Delta}_M.
  In this paper, we study the category of L-algebras (dual of L-coalgebras),
prove that the free L-algebra on one generator is constructed over rooted
planar symmetric ternary trees with odd numbers of nodes and the L-operad is
Koszul. We then introduce triplicial-algebras: vector spaces equipped with
three associative operations verifying three entanglement relations. The free
triplicial-algebra is computed and turns out to be related to even trees. Via a
general structure theorem (\`a la Cartier-Milnor-Moore), we obtain that the
category of L-algebras is equivalent to a much more structured category called
connected coassociative triplicial-bialgebras (coproduct linked to operations
via infinitesimal relations), that is the triple of operads (As, Trip, L) is
good. Bidirected graphs, related to NAP-algebras, are briefly evoked and
postponed to another paper.
