I am studying the basics of renormalisation. In 10.3 of Weinberg QFT book, motivated by the renomalisation of field, he express the bare Lagrangian as two part

\(\mathscr{L=L}_0+\mathscr{L}_1\),

\(\mathscr{L}_0=-\frac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{2}m^2\Phi^2\),

\(\mathscr{L}_1=-\frac{1}{2}(Z-1)(\partial_{\mu}\Phi\partial^{\mu}\Phi+m^2\Phi^2)+\frac{1}{2}Z\delta m^2\Phi^2-V(\Phi)\),

where

\(V(\Phi)\equiv V(\sqrt{Z}\Phi_B)\)

After expressing the corrected propagator as a geometric series of uncorrected propagator times the sum of one-particle-irreducible graphs $\Pi^*$, the only thing need to calculate is the $\Pi^*$, but he states:

In calculating $\Pi^*$, we encounter a tree graph arising from a single insertion of vertices corresponding to the terms in $\mathscr{L}_1$ proportional to $ \partial_{\mu}\Phi\partial^{\mu}\Phi$ and $\Phi^2$, plus a term $\Pi^*_{LOOP}$ arising from loop graphs like that in Figure 10.4(a):

\(\Pi^*(q^2)=-(Z-1)[q^2+m^2]+Z\delta m^2+\Pi^*_{LOOP}(q^2).\)

*What's the tree graph arising from a single insertion of vertices that corresponds to the terms in $\mathscr{L}_1$ ? And why they are proportional?*