  In [Aldous,Pitman,1998] a tree-valued Markov chain is derived by pruning off
more and more subtrees along the edges of a Galton-Watson tree. More recently,
in [Abraham,Delmas,2012], a continuous analogue of the tree-valued pruning
dynamics is constructed along L\'evy trees. In the present paper, we provide a
new topology which allows to link the discrete and the continuous dynamics by
considering them as instances of the same strong Markov process with different
initial conditions. We construct this pruning process on the space of so-called
bi-measure trees, which are metric measure spaces with an additional pruning
measure. The pruning measure is assumed to be finite on finite trees, but not
necessarily locally finite. We also characterize the pruning process
analytically via its Markovian generator and show that it is continuous in the
initial bi-measure tree. A series of examples is given, which include the
finite variance offspring case where the pruning measure is the length measure
on the underlying tree.
