  The phase transition of $M$-digging random on a general tree was studied by
Collevecchio, Huynh and Kious (2018). In this paper, we study particularly the
critical $M$-digging random walk on a superperiodic tree that is proved to be
recurrent. We keep using the techniques introduced by Collevecchio, Kious and
Sidoravicius (2017) with the aim of investigating the phase transition of
Once-excited random walk on general trees. In addition, we prove if $\mathcal
T$ is a tree whose branching number is larger than $1$, any multi-excited
random walk on $\mathcal{T}$ moving, after excitation, like a simple random
walk is transient.
