A bond incident degree (BID) index $TI(G)$ of a connected graph $G$ with edge-weight function $I(x,y)$ is defined as $$TI(G)=\sum _{uv\in E(G)}I(d_G(u),d_G(v)),$$ where $I(x,y)>0$ is a symmetric real function with $x\ge 1$ and $y\ge 1$, $d_G(u)$ is the degree of vertex $u$ in $G$.

In this paper, we use a unified method to characterize the first two maximum and the first two minimum trees with respect to (exponential) BID indices, respectively. As corollaries, we deduce a number of previously established results, and state a few new. The results extend some results of Zeng et al. (2021) and Yang et al. (2023).