Weighted rooted trees and deformations of operads

We will define an operad \(\mathcal{B}^0\) on planar rooted trees. \(\mathcal{B}^0\) is analgous to the \(NAP\)-operad in the non-planar tree setting. We will define a family of "current-preserving" operads \(\mathcal{B}^\lambda\) depending on a scalar parameter \(\lambda\), which can be seen as a deformation of the operad \(\mathcal{B}^0\). Forgetting the extra "current preserving" notion above give back the Brace operad for \(\lambda=1\) and the \(\mathcal{B}^0\) operad for \(\lambda=0\). A natural map from non-planar rooted trees to plane ones gives back the current-preserving interpolation between \(NAP\) and pre-Lie investigated in a previous article.