Despite many very recent advances concerning the fundamental problem of phyllotaxis, to explain why the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, …) arises in the secondary spirals on plants, the domain of phyllotaxis is still seeking a rational explanation. All the known theories (Veen, Adler, Thornley, Richards, …: inhibitor, space-filling, contact pressure, mechanistic, …) try to predict the place of each primordium in the apex, on the basis of chemical or mechanical forces or fields; they all have their obvious exceptions. To take into account all the forces and gradients involved in the distal zone of the apex, from where the primordia are emerging, we must consider the problem as phylogenetic (evolution of species). We thus introduce a concept of entropy (from the Greek, evolution) in the domain of growth, by means of the new concept of relational tree, each type of phyllotaxis being represented by a tree. The latter concept is born from theoretical considerations but it is supported by botanical works (Bolle, Berdyshev, Woodger, …) and the former concept has been inspired by the works of Collot in bio-thermodynamics. In the conceptual framework thus obtained, we have proved, as an application of the Principle of Optimal Design, that the cost (the entropy) of normal asymmetric phyllotaxis (characterized by the Fibonacci sequence) is minimal among all other types of phyllotaxis. We have presented this development using the four kinds of axioms found in axiomatic mathematical physics.