1Department of Mathematics, University of Florida, Gainesville, Florida 32601 Current address: Department of Mathematics, University of Florida, Gainesville, Florida 32611
In this paper we find recurrence relations for the asymptotic probability a vertex is protected in all Motzkin trees. We use a similar technique to calculate the probabilities for balanced vertices of rank . From this we calculate upper and lower bounds for the probability a vertex is balanced and upper and lower bounds for the expected rank of balanced vertices.
