We describe a model for tree crown extraction from aerial images, a problem of great practical importance for the forestry industry. The novelty lies in the prior model of the region occupied by tree crowns in the image, which is a phase field version of the higher-order active contour inflection point `gas of circles' model. The model combines the strengths of the inflection point model with those of the phase field framework: it removes the `phantom circles' produced by the original `gas of circles' model, while executing two orders of magnitude faster than the contour-based inflection point model. The model has many other areas of application e.g., to imagery in nanotechnology, biology, and physics.