Consider a rooted binary tree with n nodes. Assign with the root the abscissa
0, and with the left (resp. right) child of a node of abscissa i the abscissa
i-1 (resp. i+1). We prove that the number of binary trees of size n having
exactly n_i nodes at abscissa i, for l =< i =< r (with n = sum_i n_i), is nlnrn0(n0−1n−1+n1)i=0l≤i≤r∏(ni−1ni−1+ni+1−1), with n_{l-1}=n_{r+1}=0. The
sequence (n_l, ..., n_{-1};n_0, ..., n_r) is called the vertical profile of the
tree. The vertical profile of a uniform random tree of size n is known to
converge, in a certain sense and after normalization, to a random mesure called
the integrated superbrownian excursion, which motivates our interest in the
profile. We prove similar looking formulas for other families of trees whose
nodes are embedded in Z. We also refine these formulas by taking into account
the number of nodes at abscissa j whose parent lies at abscissa i, and/or the
number of vertices at abscissa i having a prescribed number of children at
abscissa j, for all i and j. Our proofs are bijective.Comment: 47 page