We introduce the notion of quasi-full rooted trees. We provide a poset matrix based approach for the recognition of quasi-full rooted trees. We obtain an exact enumeration of the unlabeled quasi-full t-ary rooted trees with n elements (nodes), where 2 ≤ t < n. Here, we give the enumeration of the unlabeled quasi-full rooted trees according to the arity and the number of elements of the trees. This method induces a polynomial-time algorithm that determines the values of the parameters involved in the enumeration formula. We also computationally implement the enumeration algorithm and include the number of n-element unlabeled quasi-full t-ary rooted trees for all 2 ≤ t < n ≤ 31.