  Plucking polynomial for plane rooted trees was introduced by J.H. Przytycki
in 2014. As it was shown later, this polynomial can be used to find
coefficients $C(A)$ of Catalan states $C$ of $m \times n$-lattice crossing
$L(m,n)$ without returns on one side. In this paper, we show that $C(A)$ for
any $C$ can be found by using $\Theta_{A}$-state expansion which represents
$C(A)$ as a linear combination of coefficients of Catalan states with no top
returns over $\mathbb{Q}(A)$. We also provide an algorithm for finding
$\Theta_{A}$-state expansions and examples of its applications. Finally, an
example of a Catalan state with non-unimodal coefficient is given.
