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