  For a graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap process on $G$
is the process which starts with $G$ and, at every time step, adds any missing
edges on the vertices of $G$ that complete a copy of $H$. This process
eventually stabilises and we are interested in the extremal question raised by
Bollob\'as, of determining the maximum running time (number of time steps
before stabilising) of this process, over all possible choices of $n$-vertex
graph $G$. In this paper, we initiate a systematic study of the asymptotics of
this parameter, denoted $M_H(n)$, and its dependence on properties of the graph
$H$. Our focus is on $H$ which define relatively fast bootstrap processes, that
is, with $M_H(n)$ being at most linear in $n$. We study the graph class of
trees, showing that one can bound $M_T(n)$ by a quadratic function in $v(T)$
for all trees $T$ and all $n$. We then go on to explore the relationship
between the running time of the $H$-process and the minimum vertex degree and
connectivity of $H$.
