  In this paper, we consider various classes of polyiamonds that are animals
residing on the triangular lattice. By careful analyses through certain
layer-by-layer decompositions and cell pruning/growing arguments, we derive
explicit forms for the generating functions of the number of nonempty
translation-invariant baryiamonds (bargraphs in the triangular lattice),
column-convex polyiamonds, and convex polyiamonds with respect to their
perimeter. In particular, we show that the number of
  (A) baryiamonds of perimeter $n$ is asymptotically
$$\frac{(\xi+1)^2\sqrt{\xi^4+\xi^3-2\xi+1}}{2\sqrt{\pi n^3}}\xi^{-n-2},$$ where
$\xi$ is a root of a certain explicit polynomial of degree 5.
  (B) column-convex polyiamonds of perimeter $n$ is asymptotic to
$$\frac{(17997809\sqrt{17}+3^3\cdot13\cdot175463)\sqrt{95\sqrt{17}-119}}{2^7\cdot43^2\cdot
89^2\sqrt{6\pi n^3}}\left(\frac{3+\sqrt{17}}{2}\right)^{n-1}.$$
  (C) convex polyiamonds of perimeter $n$ is asymptotic to
$$\frac{1280}{441\sqrt{3\pi n^3}}3^n.$$
