The introduction of the spatial Lambda-Fleming-Viot model (ΛV) in population genetics was mainly driven by the pioneering work of Alison Etheridge, in collaboration with Nick Barton and Amandine Véber about ten years ago [1, 2]. The ΛV model provides a sound mathematical framework for describing the evolution of a population of related individuals along a spatial continuum. It alleviates the "pain in the torus" issue with Wright and Malécot's isolation by distance model and is sampling consistent, making it a tool of choice for statistical inference. Yet, little is known about the potential connections between the ΛV and other stochastic processes generating trees and the spatial coordinates along the corresponding lineages. This work focuses on a version of the ΛV whereby lineages move infinitely rapidly over infinitely small distances. Using simulations, we show that the induced ΛV tree-generating process is well approximated by a birth-death model. Our results also indicate that Brownian motions modelling the movements of lineages along birth-death trees do not generally provide a good approximation of the ΛV due to habitat boundaries effects that play an increasingly important role in the long run. Finally, we describe efficient algorithms for fast simulation of the backward and forward in time versions of the ΛV model.