Rotating random trees with Skorokhod’s M1 topology

We extend the classical coding of measured R-trees by continuous excursion-type functions to càdlàg excursion-type functions through the notion of parametric representations. The main feature of this extension is its continuity properties with respect to the Gromov-Hausdorff-Prokhorov topology for R-trees and Skorokhod's M₁ topology for càdlàg functions. As a first application, we study the R-trees T_x(α) encoded by excursions of spectrally positive $α$-stable Lévy processes for α∈ (1,2]. In a second time, we use this setting to study the large-scale effects of a well-known bijection between plane trees and binary trees, the so-called rotation. Marckert has proved that the rotation acts as a dilation on large uniform trees, and we show that this remains true when the rotation is applied to large critical Bienaymé trees with offspring distribution attracted to a Gaussian distribution. However, this does not hold anymore when the offspring distribution falls in the domain of attraction of an $α$-stable law with α∈ (1,2), and instead we prove that the scaling limit of the rotated trees is T_x(α).