Reals in the Matet and Willow Models

Abstract: In this article, we try to complete the regularity implications between the regularitites of the well-known tree forcing notions at the $\boldsymbol{\Delta}^1_2$ level of the projective hierarchy. The missing links in this case were the regularities corresponding to Matet and Willow tree Forcings. Some of the techniques yield more general results related to locally countable closed graphs too, which we mention as corollaries.