Embedding mapping class groups into products of trees

We prove that quasi-trees of spaces satisfying the axiomatisation given 
by Bestvina, Bromberg and Fujiwara are quasi-isometric to tree-graded 
spaces in the sense of Dru\c{t}u and Sapir. As a corollary we deduce 
that mapping class groups quasi-isometrically embed into a finite 
product of tree-graded spaces. This gives bounds on compression 
exponent and guarantees finite Assouad-Nagata dimension. Moreover, 
using results of Buyalo and Masur-Minsky we prove that curve complexes
embed into a finite product of trees, which we use to complete the 
theorem stated in the title."