Moreover, since [tau] is a trisection, [a.sub.3] is not the minimum inside its vertex, so the block [C.sub.3] is not empty.
We let [D.sub.g](n) = [D.sup.I.sub.g](n) [union] [D.sup.II.sub.g](n) be the set of unicellular maps of genus g with n edges, and a distinguished trisection.
The set [D.sub.1](n) of unicellular maps of genus 1 with n edges and a distinguished trisection is in bijection with the set [U.sup.3.sub.0](n) of rooted plane trees with n edges and three distinguished vertices.
Let M be a unicellular map of genus 2, and [tau] be a trisection of M.
The map (M', [tau]) is a unicellular map of genus 1 with a distinguished trisection: therefore we can apply the mapping [[PSI].sup.-1] to (M', [tau]).
The set [D.sub.2](n) of unicellular maps of genus 2 with one marked trisection is in bijection with the set [U.sup.3.sub.1](n) [union] [U.sup.5.sub.0] (n).
Glue the three last vertices [v.sub.2]q-i,[v.sub.2]q, [v.sub.2]q+1 together, via the mapping [PHI], in order to obtain a new map Mi of genus p + I with a distinguished trisection [tau] of type I.
Let ([v.sub.2]q-2i-i,[v.sub.2]q-2i, [tau]) be the triple consisting of the last two vertices which have not been used until now, and the trisection [tau].
We let A(M, v*) := (Mq, [tau]) be the map with a distinguished trisection obtained at the end of this procedure.
In other words, all unicellular maps of genus g with a distinguished trisection can be obtained in a canonical way by starting with a map of lower genus with an odd number of distinguished vertices, and then applying once the mapping and a certain number of times the mapping 'J/.