  We study the limiting degree distribution of the vertex splitting model
introduced in \cite{DDJS:2009}. This is a model of randomly growing ordered
trees, where in each time step the tree is separated into two components by
splitting a vertex into two, and then inserting an edge between the two new
vertices. Under some assumptions on the parameters, related to the growth of
the maximal degree of the tree, we prove that the vertex degree densities
converge almost surely to constants which satisfy a system of equations. Using
this we are also able to strengthen and prove some previously non-rigorous
results mentioned in the literature.
