2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72283We continue our reformulation of free dendriform algebras, dealing this time with the free dendriform trialgebra generated be Y over planar rooted trees. We propose a 'deformation' of a vectorial coding used in Part I, giving a LL-lattice on rooted planar trees according to the terminology of A. Blass and B. E. Sagan. The three main operations on trees become explicit, giving thus a complementary approach to a very recent work of P. palacios and M. Ronco. Our parenthesis framework allows a more tractable reformulation to explore the properties of the underlying lattice describing operations and simplify a proof of a fundamental theorem related to arithmetics over trees, the so-called arithmetree. Arithmetree is then viewed as a noncommutative extention of (N,+,x), the integers being played by the corollas. We give also two representations of the super Catalan numbers or Schroder numbers.19 pagesCombinatoricsQuantum Algebra05C05; 06A07; 11A99text