
146 Algebraic Operads: An Algorithmic Companion
The parent function and the planar structure on the thus defined set of vertices
are induced by the respective parent functions and planar structures of τ
i
,
0 ≤ i ≤ r, with the following exceptions. For each j = 1, . . . , r, for the only
vertex v
j
in Parent
−1
τ
j
(Root(τ
j
)), we define
Parent
τ
(v
j
) := Parent
τ
0
(
j
),
where
j
= n
−1
0
(j) is the leaf of τ
0
numbered by j. This means that
Parent
−1
τ
(Parent
τ
0
(
j
)) = {v
j
} tParent
−1
τ
0
(Parent
τ
0
(
j
)) \{
j
};
the total order needed by the planar structure puts v
j
in the place of
j
.
The labelling x of Int(τ) =
F
r
i=0
Int(τ
i
) is given by the disjoint union of
labellings x
j
, 1 ≤ j