  Consider a Markov chain on the space of rooted real binary trees that
randomly removes leaves and reinserts them on a random edge and suitably
rescales the lengths of edges. This chain was introduced by David Aldous who
conjectured a diffusion limit of this chain, as the size of the tree grows, on
the space of continuum trees. We prove the existence of a process on continuum
trees, which via a random time change, displays properties one would expect
from the conjectured Aldous diffusion. The existence of our process is proved
by considering an explicit scaled limit of a Poissonized version of the Aldous
Markov chain running on finite trees. The analysis involves taking limit of a
sequence of splitting trees whose age processes converge but the contour
process does not. Several formulas about the limiting process are derived.
