
82 Algebraic Operads: An Algorithmic Companion
one allows one to work with divisibility algorithmically. We will now explain
how the same is done for nonsymmetric operads. First of all, the notion of a
subword of a word becomes, in the case of operations, the notion of a subtree
of a given tree.
Definition 3.4.2.1 (Subtree of a planar rooted tree). Let τ be a rooted
tree. Suppose that V
0
⊂ Int(τ) is a nonempty subset satisfying the following
properties:
• there exists just one vertex v
0
∈ V
0
for which Parent
τ
(v
0
) is not in V
0
,
• for each vertex v
00
∈ V
0
there is a (unique) nonnegative integer l and
vertices v
0
= v
0
, v
1
, . . . , v
l
= v
00
, such that v
i
= Paren