  We study two families of lattices whose number of elements are given by the
numbers in even (respectively odd) positions in the Fibonacci sequence. The
even Fibonacci lattice arises as the lattice of simple elements of a Garside
monoid partially ordered by left-divisibility, and the odd Fibonacci lattice is
an order ideal in the even one. We give a combinatorial proof of the lattice
property, relying on a description of words for the Garside element in terms of
Schr\"oder trees, and on a recursive description of the even Fibonacci lattice.
This yields an explicit formula to calculate meets and joins in the lattice. As
a byproduct we also obtain that the number of words for the Garside element is
given by a little Schr\"oder number.
