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1. Understanding the general principles governing the impressive diversity of plant morphology has long been a goal of botanists. However, the broad variability of plant growth forms has challenged the development of general models of plant growth.

2. A recent theoretical model, the fractal branching model of West, Brown and Enquist (WBE), purports to explain the scaling of plant form in a variety of taxa; however, its applicability to clades that do not meet its underlying assumptions, particularly plants that lack volume-filling branching, has been unclear.

3. Here we show how an extension of the WBE model, the minimal branching model, can quantitatively predict the scaling of form in plants lacking volume-filling branching. We then test the model’s predictions with data from a biometric database on Sonoran Desert plants.

4. As predicted, empirical data support the ubiquity of the 3/4-power scaling of photosynthetic surface area in plants, but nevertheless show that the morphological dimensions (height, spread) in plants with minimal branching scale with exponents differently from those in plants with fractal-like external branching.

5. We then compare expectations under the minimal branching model with those of geometric similitude and fractal branching models, which make predictions that are close to those of the minimal model. Confidence intervals for empirical data sometimes include all three models. However, unanimous agreement in interspecific cases, and greater support in intraspecific cases, appear to favour the minimal model.

6. It is generally thought that succulents, particularly cacti, exhibit morphological adaptations that limit water loss via surface areas while increasing the capacity for water storage. Our model and supporting empirical data strongly suggest that the succulent morphology has evolved from selection to minimize external branching but not necessarily the scaling of external surface areas.

7. Our work demonstrates that a common body of allometric theory, based on the scaling of resource-exchange networks, provides a theoretical baseline that can account for much diversity in land plant form and architecture.