$t$-adic symmetrization map on harmonic algebra

Bachmann, Takeyama and Tasaka introduced a $\mathbb{Q}$-linear map $\phi$, which we call the symmetrization map in this paper, on the harmonic algebra $\mathfrak{H}^1$. They calculated $\phi(w)$ explicitly for an element $w$ in $\mathfrak{H}^1$ related to the multiple zeta values of Mordell--Tornheim type. In this paper, we introduce its $t$-adic generalization $\widehat{\phi}$ and calculate $\widehat{\phi}(w)$ for an element $w$ in $\mathfrak{H}^1[[t]]$ constructed from the theory of $2$-colored rooted tree.