Random mappings with fixed elements

We consider orgraphs of random mappings of $n$-element set having uniform distribution on the set of all mappings with fixed points. Formulas for exact and limit distributions of the number of vertices in a tree or in a forest containing a given vertex are derived along with analogous results for the height of a given vertex conditioned to belong to a forest. Exact and limit distributions of the height of a given vertex in a random rooted tree or forest with $n$ vertices are derived also.