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We consider $f:\hat I\to \R$ being a $C^3$ (or $C^2$ with bounded distortion)real-valued multimodal map with non-flat critical points, defined on $\hat I$being the union of closed intervals, and its restriction to the maximal forwardinvariant subset $K\subset I$. We assume that $f|_K$ is topologicallytransitive. We call this setting the (generalized multimodal) real case. Weconsider also $f:\C\to \C$ a rational map on the Riemann sphere and itsrestriction to $K=J(f)$ being Julia set (the complex case). We considertopological pressure $P_{\spanning}(t)$ for the potential function$\varphi_t=-t\log |f'|$ for $t>0$ and iteration of $f$ defined in a standardway using $(n,\e)$-spanning sets. Despite of $\phi_t=\infty$ at critical pointsof $f$, this definition makes sense (unlike the standard definition using$(n,\e)$-separated sets) and we prove that $P_{\spanning}(t)$ is equal to otherpressure quantities, called for this potential {\it geometric pressure}, in thereal case under mild additional assumptions, and in the complex case providedthere is at most one critical point with forward trajectory accumulating in$J(f)$. $P_{\spanning}(t)$ is proved to be finite for general rational maps,but it may occur infinite in the real case. We also prove that geometric treepressure in the real case is the same for trees rooted at all `safe' points, inparticular at all points except the set of Hausdorff dimension 0.

Geometric pressure in real and complex 1-dimensional dynamics via trees of preimages and via spanning sets