No cutoff in Spherically symmetric trees

We show that for lazy simple random walks on finite spherically symmetric trees, the ratio of mixing time and relaxation is bounded by a universal constant. Consequently, any sequence trees do not exhibit pre-cutoff; this conclusion also holds continuous-time walks. This answers question recently proposed Gantert, Nestoridi, Schmid. hitting times vertices are (uniformly) non concentrated. Finally, we study stability our results under rough isometries.