Planar packing of trees and spider trees

In [A. García, C. Hernando, F. Hurtado, M. Noy, J. Tejel, Packing trees into planar graphs, J. Graph Theory (2002) 172–181] García et al. conjectured that for every two non-star trees there exists a planar graph containing them as edge-disjoint subgraphs. In this paper we prove the conjecture in the case in which one of the trees is a spider tree.