Suppose we have two planar graphs $G_1$ and $G_2$ with number of spanning tree count $P_1$ and $P_2$ respectively then there is an easy construction which gives a planar graph with spanning tree count $P_1P_2$ (introduce one additional vertex $v$ and join any one vertex in Graph $G_1$ to the vertex $v$ and any one vertex in Graph $G_2$ to the vertex $v$) and this contruction does not need to know $P_1$ and $P_2$.

Is there a graph construction in polynomial time to get a planar graph from $G_1$ and $G_2$ with spanning tree count $P_1+P_2$ without knowing $P_1$ or $P_2$?