... denote the set of standard Catalan trees by standard(C) and the set of standard marked Catalan trees by standard(C, ∗) The standard trees in C4 are , , , , One of the standard trees in CH is The ... x then write w = a1 W and w = b1 W Then wx = a1 W x and w x = b1 W x If a1 > b1 then clearly wx > w x On the other hand, if a1 = b1 then W ≥ W , hence by the induction hypothesis W x ≥ W x, and ... Tree Expansion of the Inverse Catalan trees are rooted planar trees whose internal vertices have out-degree ≥ We will denote the set of Catalan trees by C and the set of Catalan trees having p external...